UNCERTAIN FUNCTIONAL DIFFERENTIAL

EQUATIONS

Name : Umber Rana

Year of Admission : 2008

Registration No. : 115-GCU-PHD-SMS-08

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

i

UNCERTAIN FUNCTIONAL DIFFERENTIAL

EQUATIONS

Submitted to

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

in the partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy

in

Mathematics

By

Name : Umber Rana

Year of Admission : 2008

Registration No. : 115-GCU-PHD-SMS-08

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

ii

DECLARATION

I, Ms Umber Rana Registration No. 115-GCU-PHD-SMS-08 student at Abdus

Salam School of Mathematical Sciences GC University in the subject of

Mathematics year of admission 2008, hereby declare that the matter printed in

this thesis titled

“UNCERTAIN FUNCTIONAL DIFFERENTIAL EQUATIONS”

is my own work and that

(i) I am not registered for the similar degree elsewhere contemporaneously.

(ii) No direct major work had already been done by me or anybody else on

this topic; I worked on, for the Ph. D. degree.

(iii) The work, I am submitting for the Ph. D. degree has not already been

submitted elsewhere and shall not in future be submitted by me for

obtaining similar degree from any other institution.

Dated: ------------------------- ------------------------------------

Signature

iii

RESEARCH COMPLETION CERTIFICATE

Certified that the research work contained in this thesis titled

“UNCERTAIN FUNCTIONAL DIFFERENTIAL EQUATIONS”

has been carried out and completed by Ms Umber Rana Registration No.

115-GCU-PHD-SMS-08 under my supervision.

----------------------------- -------------------------------

Date Vasile Lupulescu

Supervisor

Submitted Through

Prof. Dr. A. D. Raza Choudary --------------------------------

Director General Controller of Examination

Abdus Salam School of Mathematical Sciences GC University Lahore

GC University Lahore Pakistan.

Pakistan.

To my respectable parents

&

husband

iv

Table of Contents

Table of Contents v

Acknowledgements vii

Introduction 1

0.1 Personal Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Set functional differential equations in Banach spaces 4

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Set functional differential equations . . . . . . . . . . . . . . . . . . . 12

1.4 Continuous dependence results . . . . . . . . . . . . . . . . . . . . . . 19

2 Neutral set differential equations 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Existence and Uniqueness set functional differential equation of neutral

type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Continuous Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Fuzzy delay differential equations 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Fuzzy delay differential equation . . . . . . . . . . . . . . . . . . . . . 42

4 Uncertain dynamic systems on time scales 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Uncertain process on time scales: . . . . . . . . . . . . . . . . 56

v

4.3 Uncertain initial value problem . . . . . . . . . . . . . . . . . . . . . 57

4.4 Uncertain linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 63

Conclusion 72

Bibliography 73

vi

Acknowledgements

Praise and glory be to You, O Allah. Blessed be Your Name, exalted be Your Majesty

and Glory. There is no god but You.

Only with Allah’s blessing I got a lot of help, support and encouragement from

many people so that I could finish writing this thesis. For that, I would like to

sincerely give my praises to Allah and to express my gratitude to all the people who

helped me in accomplishing this thesis.

First of all, I would like to thank my supervisor, Prof. Dr. Vasile Lupulescu ,

for his many creative guidance, continuous support, encouragement and patience in

guiding me during the research. I appreciate all his contributions of time and ideas to

make my Ph.D. experience productive and stimulating. Secondly, my special thanks

go to Professor A. D. R. Choudary (Director General ASSMS) for his support and

guidance through the tough times at the start of my candidature.

Next, I pay my gratitude to all foreign faculty at ASSMS for improving my skills

in Mathematics.

I am thankful to the HEC for awarding me the Indigenous Fellowship and also

thankful to Government of Punjab for providing me financial support and such a

good research circumstances.

I also deeply thank to all the official staff in ASSMS, in particular, Mr. Awais,

Mr. Numan, Mr. Shaokat, Javaid.

Of course, at this stage I cannot forget my whole family especially my respectable

parents, parents in law and my husband Dr. Muntazim Abbas Hashmi for their

prayers and continuing support. Their prayers, have lightened up my spirit to finish

this thesis.

Umber Abbas Hashmi

ASSMS, GC University, Lahore.

vii

Introduction

The nature of a dynamic system in engineering or natural sciences depends on the

accuracy of the information we have about the parameters that describe that system.

If our knowledge about a dynamic system are precise then this is a deterministic

dynamical system. A common mathematical model used to study the behavior of de-

terministic dynamical systems is the ordinary differential equations. Unfortunately,

our knowledge about the parameters of a dynamic system are accurate only in very

rare cases, because in most cases the available data for the description and evaluation

of the parameters are inaccurate, imprecise or confusing. In other words, evalua-

tion of parameters of a dynamical system is not without uncertainties. When our

knowledge about the parameters of a dynamic system are of statistical nature, that

is, the information is probabilistic, the common approach in mathematical modeling

of such systems are random differential equations or stochastic differential equations.

Statistical information are not always appropriate or they can be obtained quite dif-

ficult. Recent studies have shown that there are many types of uncertainties on the

evaluation parameters or functional relations initial conditions of a dynamic system.

The theory of fuzzy sets has numerous applications in modeling many real-world phe-

nomena such as pattern recognition, decision-making under uncertainty, information

retrieval, large-systems control, management science, artificial neural systems, to give

only few examples. This theory tries and succeeds in large part to give an adequate

answer about the possibility of modeling the natural processes with insuficient data

or uncertain data. Notion of fuzzy sets was introduced by Lofti Zadeh ([150]) via

membership function. Fuzzy sets were defined as a model for inexact concepts and

subjective judgments. Such a model can be used in situations when deterministic

and/or probabilistic models do not provide a realistic description of the process un-

der study. The examples given by Zadeh in his article were more abstract, fuzzy set

theory has proved its usefulness in practice over time. Fuzzy differential equations

are an appropriate tool in modeling of dynamic systems for which the parameters

1

2

have ambiguous or vague values. It is well known that there is a strong connection

between fuzzy differential equations and set differential equations [86, 88, 105, 43].

The fuzzy differential equations can be also interpreted in terms of differential in-

clusions. The differential inclusions seems to be better suited to modeling practical

situations under uncertainty and imprecision. Choosing a mathematical model for

modeling dynamic systems with uncertainty depends on the types of uncertainty and

the nature of the system. There is no perfect and universal mathematical model for

modeling of dynamic systems with uncertainty. Each of the proposed mathematical

model has some shortcomings and drawbacks. This observation is also true for models

using the concepts of fuzzy sets and fuzzy differential equations. Another approach

to study the behavior of fuzzy phenomena was proposed by Liu ([92, 96, 95, 93])

using the notion of credibility measure. Credibility theory is based on five axioms

from which a credibility measure is defined. As extensions of credibility theory, Liu

[96] introduced the notiuon of fuzzy variable as a function from a credibility space to

the set of real numbers and the notion of fuzzy process as a family of fuzzy variables.

Fuzzy differential equation was proposed by Li and Liu ([99, 92, 109]) as a type of

differential equation driven by Liu process just like that stochastic differential equa-

tions driven by Brownian motion. Credibility theory began to play an important role

in mathematical modeling of dynamic systems with more uncertainty coming from

biology, engineering, physics and other sciences. Some applications of credibility the-

ory, including fuzzy optimization, scheduling, engineering design, portfolio selection,

capital budgeting, vehicle routing, production planning, reliability can be find in

[9, 20, 85, 113, 122, 127]. However, many surveys showed that there exist imprecise

quantities which cannot be quantified by credibility measure and then they are not

fuzzy concepts. This means that some real problems cannot be modeled by credibility

theory. In order to overcome this shortcoming, Liu [93] founded an uncertainty theory

that is a branch of mathematics based on normality, monotonicity, self-duality, and

countable subadditivity axioms. The concept of uncertain differential equation was

also introduced in [93, 52]. In this work we study of the existence and uniqueness of

solutions of certain classes of differential equations that arise in modeling dynamic

systems with uncertainty.

0.1 Personal Contribution

The thesis is based on the following papers:

3

[1 ] Umber Abbas and Vasile Lupulescu, Set functional differential equations in

Banach spaces, Communications on Applied Nonlinear Analysis 18(1)(2011)

97-110.

[2 ]Umber Abbas, Vasile Lupulescu, Donal O’Regan, Awais Younus, Neutral Set

Differential Equations, Czechoslovakia Mathematical Journal (Submitted).

[3 ]Vasile Lupulescu and Umber Abbas, Fuzzy delay differential equations, Fuzzy

Optimization and Decision Making , 11(1)(2012), 99-111 (ISI Journal- impact

factor:(1.488)

[4 ] Umber Abbas, Vasile Lupulescu, Ghaus ur Rahman, Uncertain Dynamic Sys-

tems on time Scales, Journal of Uncertain Systems, (Accepted)

In the paper 1, we establish the existence of solutions to the Cauchy problem for

set differential equations with causal operator. In the last section we establish the

continuous dependence of the solutions set on initial data.

The aim of the paper 2 is to establish an existence and uniqueness result for a class

of set functional differential equation of neutral type. The continuous dependence of

solutions on initial data and parameters is also studied.

In paper 3, we prove a local existence and uniqueness result for fuzzy delay dif-

ferential equations driven by Liu process. We also establish continuous dependence

of solution with respect to initial data.

In the paper 4, we study the existence and uniqueness of the solution for the dy-

namic systems on time scales with uncertain parameters. For this aim, we introduced

the notion of uncertain process on time scales. The linear dynamic systems on time

scales with uncertain parameters are also studied.

Chapter 1

Set functional differentialequations in Banach spaces

1.1 Introduction

The set differential equations generated by differential equations when the needed

convexity is missing have been introduced in a semi-linear metric space, consisting

of all nonempty, compact and convex subsets of an initial finite or infinite dimen-

sional space. The basic existence and uniqueness results of such set differential

equations have been investigated and their solutions have compact, convex values.

The study of the theory of set differential equations as an independent discipline,

has certain advantages. For example, when the set is a single valued function, it

clear that the Hukuhara derivative and the integral utilized in formulating the set

differential equations reduce to the ordinary derivative and the integral, and there-

fore, the results obtained in the framework of set differential equations be come

the corresponding results of ordinary differential equations. Also, one can utilize

the set differential equations to investigate fuzzy differential equations. The study

of functional equations with causal operators has a rapid development in the last

years and some results are assembled in a recent monograph [94]. The term of

4

5

causal operators is adopted from engineering literature and the theory these oper-

ators has the powerful quality of unifying ordinary differential equations, integro-

differential equations, differential equations with finite or infinite delay, Volterra inte-

gral equations, and neutral functional equations, to name a few. For results, references

and applications in this framework we refer to the papers [10, 114, 115, 118, 127].

The study of set differential equations in a suitable space was initiated as an in-

dependent subject and several basic results on existence, uniqueness, comparison

result, global existence and continuous dependence are discussed in many papers

[7, 16, 25, 27, 29, 38, 46, 21, 51, 22, 23, 19, 71, 75, 126, 77, 80, 81, 20, 82, 83, 85, 87, 116,

117, 118, 121, 122, 123, 125, 126, 127, 128, 132, 136, 137, 138, 139, 142, 44, 144, 145].

For results, references and applications in this framework we refer to the book by

V. Lakshmikantham, T. Gnana Bhaskar and J. Vasundhara Devi ([84]). The aim of

this chapter is to established the existence of solutions and some properties of set

solutions for set differential equations with causal operator.

1.2 Preliminary

Let E be a real separable Banach space. We introduced the following notations:

Pb(E) the family of all nonempty bounded subsets of E,

C(E) the family of all nonempty compact subsets of E,

Cb(E) the family of all nonempty closed and bounded subsets of E,

Cc(E) the family of all nonempty closed and convex subsets of E,

Cbc(E) the family of all nonempty closed, convex and bounded subsets of E,

Kc(E) the family of all nonempty compact and convex subsets of E.

The Hausdorff-Pompeiu metric D on C(E) is defined by

D(A,B) = max

supx∈A

infy∈B||x− y||, sup

y∈Binfx∈A||x− y||

,

6

where || · || is a norm on E. It is known [146] that (C(E), D) is a complete metric

space, and Cbc(E) and Kc(E) are closed subsets of C(E). Moreover, (Kc(E), D) is

a complete and separable metric space with respect to D. The Hausdorff distance

satisfies the following properties:

D[A+ C,B + C] ≤ D[A,B] and D[A,B] = D[B,A],

D[λA, λB] = λD[A,B],

D[A,B] ≤ D[A,C] +D[C,B].

(1.1)

for all A,B,C ∈ C(E) and λ ∈ R+.

Let A,B ⊂ E. The set C ⊂ E satisfying A = B + C is known as the geometric

difference of the sets A and B and is denoted by A−B.

Let I be any bounded interval in R. We say that the set-valued mapping F : I →

Kc(E) has a Hukuhara derivative at a point t0 ∈ I if there exists DHF (t0) ∈ Kc(E)

such that the limits

limh→0+

F (t0 + h)− F (t0)

hand lim

h→0+

F (t0)− F (t0 − h)

h

exist in the topology of Kc(E) and are equal to DHF (t0). In this definition, we assume

that both differences F (t0 + h) − F (t0) and F (t0) − F (t0 − h) exist for sufficiently

small h > 0 such that t0 + h and t0 − h both belong to I.

Let us give some results in the theory of measurable multifunctions. Let T = [0, a]

be a segment of R+ with the Lebesgue measure h. We denote by M(T,C(E)) of all

measurable multifunctions from T to C(E), that is, the measurable multifunction

is a multifunction F : T → C(E) such that F−(B) = t ∈ T ;F (t) ∩ B 6= ∅ is

a measurable set for any closed set B ⊂ E. Let 1 ≤ p < ∞. A multifunction

F : T → C(E) is called Lp−integrably (essentially) bounded on T if there is h ∈

Lp(T,R+) (M > 0) such that D[F (t), θ] ≤ h(t) (D[F (t), θ] ≤ M) a.e. on T . It

7

is known [59] that F ∈ M(T,C(E)) is Lp−integrably bounded if and only if the

function t 7→ D(F (t), θ) from T to R+ belong to Lp(T,R+). Also, it easy to see that

F ∈M(T,C(E)) is essentially bounded if and only if the function t 7→ D[F (t), θ] from

T to R+ belong to L∞(T,R+). For 1 ≤ p ≤ ∞ denote by Lp(T,C(E)) the space of

all Lp−integrably multifunctions bounded inM(T,Cb(E)), where two multifunctions

F,G ∈ Lp(T,C(E)) are considered to be identical if F (t) = G(t) a.e. on T . Also, we

denote by L∞(T,Cb(E)) the space of all essentially bounded multifunctions bounded

in M(T,Cb(E)), where two multifunctions F,G ∈ Lp(T,Cb(E)) are considered to be

identical if F (t) = G(t) a.e. on T .

Since D[F (t), G(t)] ≤ D[F (t), θ] +D[G(t), θ], the function t 7→ D[F (t), G(t)] is in

Lp(T,C(E)). Therefore, if F,G ∈ Lp(T,C(E)), 1 ≤ p ≤ ∞,we can define

Dp(F,G) =

(∫T

Dp[F (t), G(t)]dt

) 1p

for 1 ≤ p <∞

and

D∞(F,G) = ess supt∈T

D[F (t), G(t)] := infM > 0;D[F (t), G(t)] < M a.e.on T.

We define two subspaces of Lp(T,C(E)), 1 ≤ p ≤ ∞, as follows

Lp(T,Cbc(E)) = F ∈ Lp(T,Cb(E));F (t) ∈ Cb

c(E) a.e.on T,

Lp(T,Kc(E)) = F ∈ Lp(T,Cb(E));F (t) ∈ Kc(E) a.e.on T.

It is known [105] that Lp(T,C(E)), 1 ≤ p < ∞, is a complete metric space with

respect to the metric Dp and Lp(T,Kc(E)) ⊂ Lp(T,Cbc(E)) are closed subspaces of

Lp(T,C(E)).

Theorem 1.1. L∞(T,C(E)), is a complete metric space with respect to the metricD∞ and L∞(T,Kc(E)) ⊂ L∞(T,Cb

c(E)) is a closed subspace of L∞(T,C(E)).

8

Proof. Let Xnn≥1 ⊂ L∞(T,C(E)) be a Cauchy sequence. We have to show thatthere is some X ∈ L∞(T,C(E)) such that limn→∞D∞(Xn, X) = 0. Since Xnn≥1 isa Cauchy sequence, then for each ε > 0 there is nε ∈ N such that

D[Xn(t), Xm(t)] ≤ D∞(Xn, X) < ε, a.e. on T and all n,m ≥ nε.

It follows that there is a null set A ⊂ T such that D[Xn(t), Xm(t)] < ε for all t ∈ TAand all n,m ≥ nε. But Xn(t)n≥1 is a Cauchy sequence in C(E), hence there isX(t) ∈ C(E) such that limn→∞D[Xn(t), X(t)] = 0 for all t ∈ TA, and moreover,X is a measurable and essentially bounded multifunction, that is X ∈ L∞(T,C(E)).Since

D[Xn(t), X(t)] = limm→∞

D[Xn(t), Xm(t)] < ε

for all t ∈ TA and all n ≥ nε, it follows that D∞(Xn, X) < ε and all n ≥ nε. Thisshow that limn→∞D∞(Xn, X) = 0. For the second part of the proposition it is trivialthat L∞(T,Kc(E)) ⊂ L∞(T,C(E)) and the standard method applies to give that thelatter is a closed subset of L∞(T,C(E)).

Remark 1. By standard method we can show that, if 1 ≤ p < q ≤ ∞, thenLq(T,C(E)) ⊂ Lp(T,C(E)), Lq(T,Cb

c(E)) ⊂ Lp(T,Cbc(E)) and Lq(T,Kc(E)) ⊂

Lp(T,Kc(E)). Also, if F,G ∈ L∞(T,Cb(E)) then

D∞(F,G) = limp→∞

(∫T

Dp[F (t), G(t)]dt

) 1p

.

Theorem 1.2. Let Xnn≥1 be a convergent sequence in L∞(T,C(E)). Then Xn

converges to X in L∞(T,C(E)) if and only if there is a null set T0 ⊂ T such that Xn

converges uniformly to X on TT0 with respect to the metric D.

Proof. If Xn converges uniformly to X on TT0 with h(T0) = 0, then for each ε > 0there is nε ∈ N such that

supt∈TT0

D[Xn(t), X(t)] < ε for all n ≥ nε.

It follows thatD[Xn(t), X(t)] < ε for all TT0 and all n ≥ nε, and soD[Xn(t), X(t)] <ε for a.e. t ∈ T and all n ≥ nε. Then we obtain that

D∞(Xn, X) = ess supt∈T

D[Xn(t), X(t)] < ε for all n ≥ nε,

and so Xn converges to X in L∞(T,C(E)).Conversely, if Xn converges to X in L∞(T,C(E)), then for each ε > 0 there is

nε ∈ N such thatess sup

t∈TD[Xn(t), X(t)] < ε for all n ≥ nε.

9

It follows that, for each n ≥ nε there is Tn ⊂ T with h(Tn) = 0 such that

D[Xn(t), X(t)] < ε for all t ∈ TTn.

If T0 =⋃n≥nε

Tn then h(T0) = 0 and

D[Xn(t), X(t)] < ε for all n ≥ nε and all t ∈ TT0,

that is, supt∈TT0

D[Xn(t), X(t)] < ε for all n ≥ nε. Therefore, Xn converges uniformly

to X on TT0 with respect to the metric D.

A multifunction F : [0, T ] → C(E) is said to Aumann integrable if SpF := f ∈

Lp([0, T ], E); f(t) ∈ F (t) a.e. 6= ∅. Then the Aumann’s integral on measurable set

T0 ⊂ [0, T ] is defined by ∫T0

F (t)dt =

∫T0

f(t)dt; f ∈ S1F

.

Since Kc(E) can be embedded as a complete cone into a corresponding Ba-

nach space E, then for a measurable mapping F : [0, T ] → Kc(E) the integral

(B)∫T0F (s)ds in the sense of Bochner is introduced in a natural way (see, [59],

[146]). If F : [0, T ] → Kc(E) is strongly measurable and integrally bounded then F

is Bochner integrable and ∫T0

F (t)dt = (B)

∫T0

F (t)dt

on measurable set T0 ⊂ [0, T ]([146]). Moreover, by embedding Kc(E) as a complete

cone into a corresponding Banach space E and taking account of the theorem on

differentiation of the Bochner integral, we find that if F : [0, T ]→ Kc(E) is Bochner

integrable and

G(t) := U0 +

∫ t

0

F (s)ds, t ∈ [0, T ], U0 ∈ Kc(E), (1.2)

10

then DHG(t) there exists a.e. on [0, T ] and the equality DHG(t) = F (t) holds a.e.

on [0, T ]. Also, if F,G : [t0, T ]→ Kc(E) are Bochner integrable, then D[F (.), G(.)] :

[0, T ]→ R is integrable and

D

[∫ t

0

F (s)ds,

∫ t

0

G(s)ds

]≤∫ t

0

D[F (s), G(s)]ds, t ∈ [0, T ].

Moreover, the space Lp(T ;Kc(E)), 1 ≤ p ≤ ∞, can be embedded naturally as

a closed convex cone in Lp(T ; E), where the embedding is isometric and the addi-

tion and the multiplication by nonnegative real numbers are preserved. Therefore

Lp(T ;Kc(E)) can be regarded as usual Banach space-valued Lp-functions.

We recall that, a mapping F : [0, T ] → Kc(E) is said to be absolutely contin-

uous if for each ε > 0, there exists δ > 0 such that, for each family (sk, tk); k =

1, 2, ..., n of disjoint open intervals in [0, T ] with∑n

k=1(tk − sk) < δ, we have∑nk=1D[F (tk), F (sk)] < ε.

If F : [0, T ] → Kc(E) is Bochner integrable and G : [0, T ] → Kc(E) is defined

by (1.2), then G is absolutely continuous, DHG(t) there exists a.e. on [0, T ], and

DHG(t) = F (t) a.e. on [0, T ] (see [100]).

For an element A ∈ Pb(E) we denote by χ(A) the Hausdorff non-compactness

measure of the set A, i.e., χ(A) is the greatest lower bounded of number r > 0 such

that might be covered by a finite number balls, whose radiuses do not exceed r.

If A,B are bounded subsets of E and A denotes the closure of A, then (see [114])

(1) χ(A) = 0 if and only if A is compact;

(2) χ(A) ≤ χ(B) if A ⊂ B;

(3) χ(A) = χ(A);

11

(4) χ(λA) = |λ|χ(A) for every λ ∈ R;

(5) χ(A+B) ≤ χ(A) + χ(B);

(6) |χ(A)− χ(B)| ≤ D[A,B];

(7) If U ⊂ C([0, T, Kc(E)) is bounded and equicontinuous, and χC is the Hausdorff

non-compactness measure in the space C([0, T ], Kc(E)), then [140]

χC(U) = χ(U([0, T ])) = supχ(U(t)); t ∈ [0, T ]

where

U(t) =⋃U(t);U ∈ U

and

U([0, T ]) =⋃U(t); t ∈ [0, T ]

.

The next lemma may be known, but the author did not find a reference to it.

Lemma 1.2.1. Let Fmm≥1 be a sequence of measurable multifunctions Fm : [0, T ]→Kc(E), m ≥ 1. If G(t) :=

⋃m≥1 Fm(t), t ∈ [0, T ], then the function t → χ(G(t)) is

strongly measurable and

χ

(∫ t

0

G(s)ds

)≤∫ t

0

χ(G(s))ds, 0 ≤ t ≤ T . (1.3)

Proof. Since Fm, m ≥ 1, are strongly measurable then t 7→ G(t) is also stronglymeasurable (see [146]). On the other hand, there exists a sequence um : [0, T ] → E,m ≥ 1, of strongly measurable selections of G such that um(t);m ≥ 1 = G(t),t ∈ [0, T ]. Moreover, by Proposition 9.2 in [42], we have

χ(G(t)) = limm→∞

limn→∞

d(Fm(t), Xn),

where Xnn≥1 is a increasing sequence of subspaces with dimXn <∞ and⋃n≥1Xn =

E. Next, let S1G be the set of all Bochner integrable selections of G. Since L1([0, T ], E)

12

is separable, then there exists a sequence vmm≥1 ⊂ S1G such that vm(t);m ≥ 1 =

S1G. By the fact that the Bochner integral is a continuous linear operator fromL1([0, T ], E) into E, we infer that∫ t

0

vm(s)ds;m ≥ 1

=

∫ t

0

G(s)ds.

Now, by the properties of the measure of non-compactness and Proposition 9.3 in[42], we have

χ

(∫ t

0

G(s)ds

)= χ

(∫ t

0

vm(s)ds;m ≥ 1

)≤∫ t

0

χ (vm(s);m ≥ 1) ds.

Since vm(t);m ≥ 1 ⊂ G(t) for t ∈ [0, T ] then, from the last inequality, we obtain(1.3).

1.3 Set functional differential equations

In the following, we consider the initial value problem for the set differential equation

DHU(t) = (QU)(t), U|[−σ,0] = Ψ ∈ Cσ, (1.4)

where Cσ = C([−σ, 0], Kc(E)) and Q : C([−σ, b), Kc(E)) → Lp([0, b], Kc(E)) is a

causal operator, that is, for each τ ∈ (0, b] and for all U, V ∈ C([−σ, b], Kc(E)) with

U|[−σ,τ ] = V|[−σ,τ ] we have (QU)(t) = (QV )(t) for a.e. t ∈ [0, τ ].

In the following we assume that the operator Q satisfies the conditions:

(h1) Q is continuous;

(h2) for each r > 0, there exists h(·) ∈ Lp([0, b],R+) such that, for all U ∈ C([−σ, b), Kc(E))

with sup−σ≤t<b

D[U(t), θ] ≤ r, we have D[(QU)(t), θ] ≤ h(t) for a.e. t ∈ [0, b];

(h3) for each bounded set U ⊂ C([−σ, b], Kc(E)) we have

χ((QU)(t)) ≤ g(t, sup0≤s≤t

χ(U(s))) for a.e. t ∈ [0, b],

where g : [0, b]× R+ → R+ is a Kamke function.

13

We recall that, by a Kamke function we mean a function g : [0, b] × R+ → R+

satisfying the Caratheodory condition (g(·, w) is a measurable function for each fixed

w ∈ R+ and g(t, ·) is continuous for each fixed t ∈ [0, b]), g(t, 0) = 0 for a.e. t ∈ [0, b],

and such that w(t) = 0 is only one solution of

w(t) ≤∫ t

0

g(s, w(s))ds

with w(t) = 0 .

By a solution of (1.4) on [−σ, T ], we mean a continuous function U : [−σ, T ] →

Kc(E)), such that U|[−σ,0] = Ψ, U is absolutely continuous on [0, T ] and whose deriva-

tive DHU in the sense of Hukuhara satisfies (1.4) for a.e. t ∈ [0, T ].

Theorem 1.3. Let Q : C([−σ, b], Kc(E)) → Lp([0, b], Kc(E)) be a causal operatorsuch that the conditions (h1) - (h3) hold. Then, for every Ψ ∈ Cσ, there exists asolution U : [−σ, T ] → Kc(E) for Cauchy problem (1.4) on some interval [−σ, T ]with T ∈ (0, b].

Proof. Let δ > 0 be any number and let r := δ + sup−σ≤t<b

D[Ψ(t), θ]. If U0 ∈

C([−σ, b], Kc(E)) denotes the function defined by

U0(t) =

Ψ(t), for t ∈ [−σ, 0]

Ψ(0), for t ∈ [0, b],

then sup0≤t<b

D[U0(t), θ] ≤ r and therefore, by (h2), we have D[(QU0)(t), θ] ≤ h(t) for

a.e. t ∈ [0, b]. We choose T ∈ (0, b] such that∫ T

0h(t)dt < δ and consider the following

set

B = U ∈ C([−σ, T ], Kc(E)); , U |[−σ,0] = Ψ, sup0≤t≤T

D[U(t), U0(t)] ≤ δ.

Next consider the integral operator P : B → C([−σ, T ], Kc(E)) given by

(PU)(t) =

Ψ(t), for t ∈ [−σ, 0)

Ψ(0) +∫ t

0(QU)(s)ds, for t ∈ [0, T ],

(1.5)

14

and we show that this operator is continuous and map the set B into itself.

First, let Um → U in B. Then, for 1 ≤ p <∞ and 1/p+ 1/q = 1, we have

sup0≤t≤T

D[(PUm)(t), (PU)(t)] = sup0≤t≤T

D[∫ t

0(QUm)(s)ds,

∫ t0(QU)(s)ds

]

≤ sup0≤t≤T

∫ t0D[(QUm)(s), (QU)(s)]ds ≤

∫ T0D[(QUm)(s), (QU)(s)]ds

≤ T 1/q(∫ T

0(D[(QUm)(s), (QU)(s)])pds

)1/p

and for p =∞ we have

sup0≤t≤T

D[(QUm)(t), (QU)(t)] ≤ Tess sup0≤t≤T

D[(QUm)(t), (QU)(t).

By (h1), in the both cases, it follows that sup0≤t≤T

D[(PUm)(t), (PU)(t)] → 0 as

m→∞. Since Um|[−σ,0] = Ψ for all m ∈ N, we deduce that P : B → C([−σ, T ], E) is

a continuous operator.

Further, we observe that if U ∈ B, then sup0≤t≤T

D[U(t), θ] < r. Hence, for every

U ∈ B, we have

sup0≤t≤T

D[(PU)(t), U0(t)] = sup0≤t≤T

D

[∫ t

0

(QU)(s)ds, θ

]≤ sup

0≤t≤T

∫ t

0

D[(QU)(s), θ]ds

≤∫ T

0

D[(QU)(s), θ]ds ≤∫ T

0

h(t)dt < δ

and thus, P (B) ⊂ B. Moreover, it follows that P (B) is uniformly bounded. Also,

P (B) is uniformly equicontinuous on [−σ, T ] since, if U ∈ B and t, s ∈ [0, T ], we have

D[(PU)(t), (PU)(s)] ≤∣∣∣∣∫ t

s

h(τ)dτ

∣∣∣∣ .Next, for each m ≥ 1, we consider the Tonelli approximations

Um(t) =

U0(t), for −σ ≤ t ≤ T/m

Ψ(0) +∫ t−T/m

0(QUm)(s)ds, for T/m ≤ t ≤ T.

15

Then, for all m ≥ 1 we have Um(·) ∈ B. Moreover, for T/m ≤ t ≤ T , we have

D[(PUm)(t), Um(t)] = D[(PUm)(t), (PUm)(t− T/m)]

= D[∫ t

0(QUm)(s)ds,

∫ t−T/m0

(QUm)(s)ds]

= D[∫ t

0(QUm)(s)ds,

∫ t0(QUm)(s)ds+

∫ t−T/mt

(QUm)(s)ds]

≤ D[∫ t

t−T/m(QUm)(s)ds, θ]≤∫ tt−T/mD[(QUm)(s), θ]ds

≤∫ tt−T/m h(s)ds,

and for 0 ≤ t ≤ T/m, we have

D[(PUm)(t), Um(t)] ≤∫ T/m

0

D[(QUm)(s), θ]ds ≤∫ T/m

0

h(s)ds.

Thus we have that sup0≤t≤T

D[(PUm)(t), (Pu)(t)]→ 0 as m→∞.

Let A = Um(·);m ≥ 1. Now, we prove that the sequence A is uniformly

equicontinuous. Let ε > 0. On the closed set [0, T ], the function t→ ξ(t) =∫ t

0h(s)ds

is uniformly continuous, and so there exists some δ > 0 such that

|ξ(t)− ξ(s)| < ε, for all t, s ∈ [0, T ] with |t− s| < ε.

Let m ≥ 1, t, s ∈ [0, T ] with |t − s| < ε. Without loss of generality, we assume

that s ≤ t. We consider three exhaustive cases. First, if 0 ≤ s ≤ t ≤ T/m, then

D[Um(t), Um(s)] = 0. Second, if 0 ≤ s ≤ T/m ≤ t ≤ T , then t− T/m < δ, and so

D[Um(t), Um(s)] = D[Um(t),Ψ(0)] ≤ ξ(t− T/m) < ε.

Third, if T/m ≤ s ≤ t ≤ T , then

D[Um(t), Um(s)] ≤ |ξ(t− T/m)− ξ(s− T/m)| < ε.

16

Since Um|[−σ,0] = Ψ for all m ≥ 1, we conclude that the sequence A is uniformly

equicontinuous. Set A(t) =⋃m≥1 Um(t) for t ∈ [−σ, T ]. Then, by (1.5) and property

(5) of χ(·), we have

χ(A(t)) ≤ χ

(∫ t

0

(QA)(s)ds

)+ χ

(∫ t

t−T/m(QA)(s)ds

),

where (QA)(t) =⋃m≥1(QUm)(t).

Note that, given ε > 0, we can find m(ε) > 0 such that∫ tt−T/m h(s)ds < ε/2 for

t ∈ [0, T ] and m ≥ m(ε). Hence we have that

χ

(∫ t

t−T/m(QUm)(s)ds;m ≥ m(ε)

)≤ 2 sup

m≥m(ε)

∫ t

t−T/mh(s)ds < ε.

Using the last inequality, we obtain that

χ(A(t)) ≤ χ

(∫ t

0

(QA)(s)ds

)Since for every t ∈ [0, T ], A(t) is bounded then, by Lemma 1.2.1 and (h3), we

have that

χ(A(t)) ≤∫ t

0

χ ((QA)(s)) ds ≤∫ t

0

g(s, sup0≤τ≤s

χ (A)(s)) ds.

Since χ(A(0)) = 0 then, by (h3), we must have that χ(A(t)) = 0 for every

t ∈ [0, T ]. Moreover, since χ(A) = sup0≤t≤T

χ(A(t)) and A|[−σ,0] = Ψ we deduce that

χ(A) = 0. Therefore, A is relatively compact subset of C([−σ, T ], Kc(E)). Then, by

Arzela-Ascoli theorem (see [110]), and extracting a subsequence if necessary, we may

assume that the sequence Um(·)m≥1 converges uniformly on [0, T ] to a continuous

function U(·) ∈ B. Therefore, since

sup0≤t≤T

D[(PU)(t), U(t)] ≤ sup0≤t≤T

D[(PU)(t), (PUm)(t)]

+ sup0≤t≤T

D[(PUm)(t), Um(t)] + sup0≤t≤T

D[Un(t), U(t)]

17

then by the fact that P is a continuous operator, we obtain that

sup0≤t≤T

D[(PU)(t), U(t)] = 0.

It follows that U(t) = (PU)(t) = U0 +∫ t

0(QU)(s)ds for every t ∈ [0, T ]. Hence

u(t) =

Ψ(t), for t ∈ [−σ, 0)

Ψ(0) +∫ t

0(QU)(s)ds, for t ∈ [0, b),

solve the Cauchy problem (1.4).

Theorem 1.4. Let Q : C([−σ, b], Kc(E)) → Lp([0, b], Kc(E)) be a causal operatorsuch that the conditions (h1) and (h2) hold. In addition, we assume that

(h4) for all Ψ ∈ Cσ, there exist τ ∈ (0, b], δ > 0, and L > 0 such that, for all

U, V ∈ C([−σ, b], Kc(E)) with U|[−σ,0] = V|[−σ,0] = Ψ and U(t), V (t) ∈ Bη(Ψ(0))

for a.e. t ∈ [0, τ ], we have

Dp[QU,QV ](τ) ≤ LD[U, V ](τ).

Then, for each Ψ ∈ Cσ, there exists a unique solution U : [−σ, T ] → Kc(E) for

Cauchy problem (1.4) on some interval [−σ, T ] with T ∈ (0, b].

Proof. Let Ψ ∈ Cσ be fixed. Then, by (h4), there exist τ ∈ (0, b], η > 0, and

L > 0 such that, for all U, V ∈ C([−σ, b], Kc(E)) with U|[−σ,0] = V|[−σ,0] = Ψ and

U(t), V (t) ∈ Bη(Ψ(0)) for a.e. t ∈ [0, τ ], we have Dp[QU,QV ](τ) ≤ LD[U, V ](τ).

Define r := η + sup−σ≤t<b

D[Ψ(t), θ]. If U0 ∈ C([−σ, b], Kc(E)) denotes the function

defined by

U0(t) =

Ψ(t), for t ∈ [−σ, 0)

Ψ(0), for t ∈ [0, b],

then sup0≤t<b

D[U0(t), θ] ≤ r and therefore, by (h2), we have D[(QU0)(t), θ] ≤ µ(t) for

a.e. t ∈ [0, b]. We choose 0 < ρ < min1, τ sufficiently small such that ρL < 1, and

18

∫ ρ0µ(t)dt < η. For a fix T ∈ (0, ρ) we define

B = U ∈ C([−σ, T ], Kc(E));U |[−σ,0] = Ψ, sup0≤t≤T

D[U(t), U0(t)] ≤ δ/2

which, when equipped with the metric ∆σ(U, V )(T ) = sup−σ≤t≤T

D[U(t), V (t)] is a com-

plete metric space. Next, we define the operator P : B → C([−σ, T ], Kc(E)) given

by

(PU)(t) =

Ψ(t), for t ∈ [−σ, 0]

Ψ(0) +∫ t

0(QU)(s)ds, for t ∈ (0, T ].

Then as in Theorem 1.1, the operator P is continuous and map the set B into

itself.

Further on, for all U, V ∈ B, we have

∆σ(PU, PV )(T ) = sup−σ≤t≤T

D[(PU)(t), (PV )(t)]

= sup−σ≤t≤T

D[

∫ t

0

(QU)(s)ds,

∫ t

0

(QV )(s)ds]

≤∫ T

0

D[(QU)(s), (QV )(s)]ds

≤ T 1/qDp(QU,QV )(T ) ≤ T 1/qLD(U, V )(T )

≤ T 1/qL∆σ(U, V )(T )

for 1 ≤ p <∞ and 1/p+ 1/q = 1, and

∆σ(PU, PV )(T ) ≤ Tess sup−σ≤t≤T

D[(PU)(t), (PV )(t)]

≤ TD∞(QU,QV )(T )

≤ TLD(U, V )(T ) ≤ TL∆σ(U, V )(T )

for p = ∞. It follows that P is a contraction on B and so, there exists an unique

U ∈ B such that P (U) = U.

19

1.4 Continuous dependence results

In the following, for a fixed Ψ ∈ Cσ and a bounded set K ⊂ Kc(E), by ST (Ψ) we

denote the set of solution U(.) of Cauchy problem (1.4) on [−σ, T ] with T ∈ (0, b]

and such that U(t) ∈ K for all t ∈ [−σ, T ].

Proposition 1.4.1. Assume that Q : C([−σ, b), Kc(E)) → L∞([0, b), Kc(E)) is acausal operator such that the condition (h1) − (h3) hold. Then, for every Ψ ∈ Cσ,ST (Ψ) is compact set in C([−σ, T ], E).

Proof. We consider a sequence Um(·)m≥1 in ST (Ψ) and we shall show that thissequence contains a subsequence which converges, uniformly on [−σ, T ], to a solutionU(·) ∈ ST (Ψ). Since K is a bounded set, then there exists r > 0 such that K ⊂Br(θ) := U ∈ Kc(E);D[U ; θ] < r. By (h2), there exists h(.) ∈ L∞([0, b];R+) suchthat D[(QU)(t); θ] ≤ h(t) for a.e. t ∈ [0, T ] and for every U(.) ∈ C([−σ;T ];Kc(E))with sup

−σ≤t≤TD[U(t); θ] < r. Since Um|[−σ,0] = Ψ, we have that Um(·)→ Ψ(·) uniformly

on [−σ, 0]. On the other hand, since Um(t) = Ψ(0) +∫ t

0(QUm)(s)ds for all t ∈ [0, T ],

we have that

D[Um(t), Um(s] ≤∣∣∣∣∫ t

s

D[(QUm)(ξ), θ]dξ

∣∣∣∣≤∣∣∣∣∫ t

s

γ(ξ)dξ

∣∣∣∣ ≤ r|t− s| for s, t ∈ [0, T ].

Therefore, Um(·)m≥1 is uniformly equicontinuous on [0, T ]. As in proof of The-orem 1.3 we can show that A = Um(·);n ≥ 1 is relatively compact subset ofC([0, T ], Kc(E)) Further, by the Ascoli-Arzela theorem ([110]) and extracting a sub-sequence if necessary, we may assume that the sequence Um(·)m≥1 converges uni-formly on [0, T ] to a continuous function U(·). If we extend U(·) to [−σ, T ] such thatU |[−σ,0] = Ψ then is clearly Um(·) → U(·) uniformly on [−σ, T ]. By the continuityof the operator Q, we have that lim

n→∞QUm = QU in L∞([0, T ], Kc(E)) and, using the

Theorem 1.2, we obtain that

limn→∞

(QUm)(t) = (QU)(t) a.e. on [0, T ].

Since D[(QUm)(ξ), θ] ≤ h(t) a.e. on [0, T ] and all m ≥ 1, by the Lebesgue dominatedconvergence theorem, we have

limn→∞

∫ t

0

(QUm)(s)ds =

∫ t

0

(QU)(s)ds for all t ∈ [0, T ].

20

It follows that U(t) = limn→∞

Um(t) = Ψ(0) +∫ t

0(QU)(s)ds for all t ∈ [0, T ] and so

U(·) ∈ ST (Ψ)

Proposition 1.4.2. Assume that Q : C([−σ, b), Kc(E)) → L∞loc([0, b), Kc(E)) is acausal operator such that the condition (h1) − (h3) hold. Then the multifunctionST : Cσ → C([−σ, T ], Kc(E)) is upper semicontinuous.

Proof. Let K be a closed set in C([−σ, T ], Kc(E)) and G = Ψ ∈ Cσ;ST (Ψ)∩K 6= ∅.We must show that G is closed in Cσ. For this, let Ψmm≥1 be a sequence in Gsuch that Ψm → Ψ on [−σ, 0]. Further, for any m ≥ 1, let Um(·) ∈ ST (Ψn) ∩ K.Then, Um = Ψm on [−σ, 0] for all m ≥ 1, and Um(t) = Ψm(0) +

∫ t0(QUm)(s)ds for all

t ∈ (0, T ]. As in proof of Proposition 1.4.1 we can show that Um(·)n≥1 convergesuniformly on [−σ, T ] to a continuous function U(·) ∈ K. Since U(t) = lim

m→∞Um(t) =

Ψ(0) +∫ t

0(QU)(s)ds for all t ∈ [0, T ], we deduce that U(·) ∈ ST (Ψ) ∩ K. This prove

that G is closed and so Ψ→ ST (Ψ) is upper semicontinuous.

Chapter 2

Neutral set differential equations

The literature on related ordinary neutral differential equations is very extensive and

we refer the reader to the book [57] for details. To our knowledge, there is no paper

on set differential equations of neutral type. Some similar results for ordinary neutral

differential equations in a finite dimensional Banach spaces have been proved in the

papers [40, 73, 74]. Importance of this work is the preciseness of the results and

explanations given in this chapter.

2.1 Introduction

The study of set differential equations as an independent subject is relatively new.

The first results in this area were obtained in [13], [24], [106]. Some recent results

of interest can be found in [30],[100], [103], [104]. For more results, references and

details we refer the reader to the book [84]. We also refer the reader to the first book

[140] devoted exclusively to the subject of set differential equations on Banach spaces

and their applications to study of differntial inclusions with nonconvex right hand.

The set differential equations with delay was studied in [1], [103] and [49]. In this

21

22

chapter we are concerned with the following set differential equation of neutral typeDHX(t) = F (t,Xt, DHXt)

X|[−r,0] = Ψ,

where F : [0, b) × C0 × L10 → Kc(E) is a given function, Kc(E) is the family of all

nonempty compact and convex subsets of a separable Banach space E, C0 denotes

the space of all continuous set-valued functions X from [−r, 0] into Kc(E), L10 is the

space of all integrably bounded set-valued functions X : [−r, 0]→ Kc(E), Ψ ∈ C0 and

DH is the Hukuhara derivative. The literature on related ordinary neutral differential

equations is very extensive and we refer the reader to the book [57] for details. To our

knowledge, there is no paper on set differential equations of neutral type. Some similar

results for ordinary neutral differential equations in a finite dimensional Banach spaces

have been proved in the papers [40], [73], and [74].

2.2 Preliminaries

In the following, E is a separable Banach space with the norm ‖·‖. We denote by

Kc(E) the family of all nonempty compact and convex subsets of E. The Hausdorff-

Pompeiu metric H on Kc(E) is defined by

H(A,B) = max

supx∈A

infy∈B||x− y||, sup

y∈Binfx∈A||x− y||

.

It is known ([17], [41]) that (Kc(E),H) is a complete and separable metric space. If

C([a, b], Kc(E)) denotes the space of all continuous set-valued functions X from [a, b]

into Kc(E), then it is well known that C([a, b], Kc(E)) is a complete and separable

metric space with respect to the metric ([62])

H[a,b](X, Y ) := supt∈[a,b]

H(X(t), Y (t)).

23

Also, let us denote by L1([a, b], Kc(E)) the space of all integrably bounded set-valued

functions X : [a, b]→ Kc(E). Then L1([a, b], Kc(E)) is a complete metric space with

respect to the metric ([41], [62])

H1,[a,b](X, Y ) =

∫ b

a

H(X(t), Y (t))dt.

Remark 2. From Theorem 1.4.5. and Theorem 2.2.5 in [91] it follows that L1([a, b], Kc(E))can be regarded as the Banach space of vector-valued Bochner integrable functions,so that L1([a, b], Kc(E)) is separable and the theory of Bochner integration can beapplied to bounded integrable set-valed functions from [a, b] into a given infinite di-mensional Banach space.

In the following, we will write C[a,b] and L1[a,b] instead of C([a, b], Kc(E)) and

L1([a, b], Kc(E)), respectively.

Let A,B ⊂ E. The set C ⊂ E satisfying A = B + C is known as the geometric

difference of the sets A and B and is denoted by A−B.

We say that the set-valued mapping X : [a, b]→ Kc(E) is Hukuhara differentiable

(or H-differentiable) at a point t0 ∈ [a, b] if there exists DHX(t0) ∈ Kc(E) such that

the limits

limh→0+

X(t0 + h)−X(t0)

hand lim

h→0+

X(t0)−X(t0 − h)

h

exist in the topology of Kc(E) and are equal to DHX(t0). In this definition, we assume

that both differences X(t0+h)−X(t0) and X(t0)−X(t0−h) exist for sufficiently small

h > 0 such that t0 +h and t0−h both belong to [a, b]. DHX(t0) ∈ Kc(E) is called the

H-derivative of X at the point t0 ∈ [a, b]. A set-valued mapping X : [a, b] → Kc(E)

is called H-differentiable on [a, b] if DHX(t) exists for each point t ∈ [a, b].At the

end points of [a, b] we consider only the one sided H-derivatives. The following three

propositions are well known (see [13], [87], [100], [140])

24

Proposition 2.2.1. If Y : [a, b]→ Kc(E) is continuous, then it is integrable on [a, b].Moreover, in this case, the set-valued function X : [a, b]→ Kc(E), defined by

X(t) := X0 +

∫ t

a

Y (s)ds, t ∈ [a, b], X0 ∈ Kc(E), (2.1)

is H-differentiable on [a, b] and DHX(t) = Y (t) for t ∈ [a, b].

Proposition 2.2.2. Let X : [a, b] → Kc(E) be H-differentiable a.e. on [a, b] andassume that DHX(t) ∈ L1([a, b], Kc(E)). Then, for any t ∈ [a, b], we have

X(t) = X(τ) +

∫ t

τ

DHX(s)ds,

for τ, t ∈ [a, b].

We recall that, a mapping X : [a, b]→ Kc(E) is said to be absolutely continuous if

for each ε > 0, there exists δ > 0 such that, for each family (sk, tk); k = 1, 2, ..., n of

disjoint open intervals in [a, b] with∑n

k=1(tk−sk) < δ, we have∑n

k=1H(X(tk), X(sk)) <

ε.

We denote by AC([a, b], Kc(E)) the space of all absolutely continuous set-valued

functions from [a, b] into Kc(E).

Proposition 2.2.3. Let X : [a, b] → Kc(E) be a integrably bounded set-valued func-tion. Then the set-valued function X : [a, b] → Kc(E), defined by (2.1) is absolutelycontinuous, DHX(t) exists a.e. on [a, b], and DHX(t) = Y (t) a.e. on [a, b].

We denote byA([a, b], Kc(E)) the set of all set-valued functionsX ∈ AC([a, b], Kc(E))

having the property that they are a.e. H-differentiable on [a, b] andDHX ∈ L1([a, b], Kc(E)).

It is easy to check that

H[a,b](X, Y ) : = H[a,b](X, Y ) +H1,[a,b](DHX,DHY )

= supt∈[a,b]

H(X(t), Y (t)) +

∫ b

a

H(DHX(t), DHY (t))dt,

is a metric on A([a, b], Kc(E)).

25

Lemma 2.2.1. A([a, b], Kc(E)) is a complete metric space with respect to the metricH[a,b].

Proof. Let Xnn≥1 be a Cauchy sequence in A([a, b], Kc(E)), i.e,

limm,n→∞

H[a,b](Xm, Xn) = 0.

Then it follows that limm,n→∞

H[a,b](Xm, Xn) = 0. Since C([a, b], Kc(E)) is a complete

metric space, then there is a continuous set-valued function X : [a, b]→ Kc(E) suchthat lim

n→∞H[a,b](Xn, X) = 0. Further, since

limm,n→∞

H1,[a,b](DHXm, DHXn) = 0,

then DHXnn≥1 is a Cauchy sequence in L1([a, b], Kc(E)). Since L1([a, b], Kc(E))is a complete metric space, then there is a continuous set-valued function Y ∈L1([a, b], Kc(E)) such that lim

n→∞H1,[a,b](DHXn, Y ) = 0. Moreover, for t ∈ [a, b] we

have

H(

t∫a

DHXn(s)ds,t∫a

Y (s)ds

)≤

t∫a

H(DHXn(s), Y (s))ds

≤ H1,[a,b](DHXn, Y )→ 0 as n→∞.Therefore, it follows that

H(X(t), X(a) +

t∫a

Y (s)ds

)≤ H(X(t), Xn(t)) +H

(Xn(t), X(a) +

t∫a

Y (s)ds

)= H(X(t), Xn(t)) +H

(Xn(a) +

t∫a

DHXn(s)ds,X(a) +t∫a

Y (s)ds

)≤ H(X(t), Xn(t)) +H(X(a), Xn(a)) +H

(t∫a

DHXn(s)ds,t∫a

Y (s)ds

)→ 0

as n→∞, and so X(t) = X(a) +t∫a

Y (s)ds for any t ∈ [a, b]. By Proposition 2.2.2 it

follows that X ∈ A([a, b], Kc(E)). Obviously, limn→∞

H[a,b](Xn, X) = 0, and the proof is

complete.

Let r > 0 be given. In the following, for any b > 0, we will write C[b], L1[b] and A[b]

instead of C([−r, b], Kc(E)), L1([−r, b], Kc(E)) and A([−r, b], Kc(E)), respectively.

Then, we write H[b], H1,[b] and H[b] instead of H[−r,b], H1,[−r,b] and H[−r,b], respectively.

26

Also, for a given t ∈ [0, b], we will write Ct, L1t and At instead of C([t− r, t], Kc(E)),

L1([t−r, t], Kc(E)) and A([t−r, t], Kc(E)), respectively. Then, we denote by Ht, H1,t

and Ht the metric on Ct, L1t and At, respectively.

If t ∈ [0, b] and X : [t−r, t]→ Kc(E) are given, then we define the set-valued function

Xt : [−r, 0]→ Kc(E) by Xt(s) = X(t+ s).

Lemma 2.2.2. If t ∈ [0, b] and X ∈ At are given, then Xt ∈ A0 and

DHXt(s) = (DHX)t (s), s ∈ [−r, 0]. (2.2)

Proof. First, we show that Xt is absolutely continuous on [−r, 0]. For this, let usremark that if (sk, tk); k = 1, 2, ..., n is an arbitrary family of disjoint open intervalsin [−r, 0], then (t + sk, t + tk); k = 1, 2, ..., n is a family of disjoint open intervalsin [t − r, t]. Since X is absolutely continuous on [t − r, t], then for each ε > 0, thereexists δ > 0 such that

n∑k=1

(tk − sk) =n∑k=1

[(t+ tk)− (t+ sk)] < δ

impliesn∑k=1

H(Xt(tk), Xt(sk)) =n∑k=1

H(X(t+ tk), X(t+ sk)) < ε,

that is, Xt is absolutely continuous on [−r, 0]. Next, we show thatXt isH-differentiableon [−r, 0] and (2.2) holds. Let s ∈ [−r, 0] be given. Since X is H-differentiable on[t− r, t] then both differences X(t + s + h)−X(t + s) and X(t + s)−X(t + s− h)exist for sufficiently small h > 0 such that t + s + h and t + s − h both belong to[t− r, t]. It follows that

limh→0+

Xt(s+ h)−Xt(s)

h= lim

h→0+

X(t+ s+ h)−X(t+ s)

h= DHX(t+ s) = (DHX)t (s),

limh→0+

Xt(s)−Xt(s− h)

h= lim

h→0+

X(t+ s)−X(t+ s− h)

h= DHX(t+ s) = (DHX)t (s),

that is, Xt is H-differentiable on [−r, 0] and (2.2) holds. Finally, we show thatDHXt ∈ L1

0. Since X ∈ At, then DHX ∈ L1t , that is, DHX is measurable and

27

integrably bounded on [t− r, t]. Therefore, there exists a sequence Xnn≥1 of simpleset-valued functions from [t− r, t] into Kc(E) such that lim

n→∞H(Xn(τ), DHX(τ)) = 0

for a.e. τ ∈ [t − r, t]. Also, it is easy to check that Xnt n≥1 is a sequence of simple

set-valued functions from [−r, 0] into Kc(E). It follows that

limn→∞H(Xn

t (s), DHXt(s)) = limn→∞H(Xn(t+ s), DHX(t+ s)) = 0

for a.e. s ∈ [−r, 0], that is, DHXt is measurable on [−r, 0]. Obviously, DHXt isintegrably bounded on [−r, 0], and thus DHXt ∈ L1

0.

Lemma 2.2.3. If t ∈ [0, b] is given, then H0(Xt, Yt) = Ht(X, Y ) for any X, Y ∈ At. Inparticular, H0(Xt, Yt) = Ht(X, Y ) for any X, Y ∈ Ct, and H1,0(Xt, Yt) = H1,t(X, Y )for any X, Y ∈ L1

t .

Proof. Indeed, we have

H0(Xt, Yt) = sups∈[−r,0]

H(Xt(s), Yt(s)) +

∫ 0

−rH(DHXt(s), DHYt(s))ds

= sups∈[−r,0]

H(X(t+ s), Y (t+ s)) +

∫ 0

−rH(DHX(t+ s), DHY (t+ s))ds

= supσ∈[t−r,t]

H(X(σ), Y (σ)) +

∫ t

t−rH(DHX(σ), DHY (σ))dσ,

that is, H0(Xt, Yt) = Ht(X, Y ).

2.3 Existence and Uniqueness set functional dif-

ferential equation of neutral type

In this section we consider the following set differential equation of neutral typeDHX(t) = F (t,Xt, DHXt)

X|[−r,0] = Ψ,(2.3)

where F : [0, b)× C0 × L10 → Kc(E) is a given function and Ψ ∈ A0. By a solution of

the initial value problem (2.3) on some interval [−r, T ], we mean a set-valued function

X ∈ AT such that X0 = Ψ and DHX(t) = F (t,Xt, DHXt) for a.e. t ∈ [0, T ].

We say that F : [0, b)× C0 × L10 → Kc(E) is a Caratheodory set-valued function if

28

(a) for a.e. t ∈ [0, b), F (t, ·, ·) is continuous,

(b) for any (U, V ) ∈ C0 × L10, F (·, U, V ) is measurable,

(c) For any bounded B ⊂ C0 × L10, there exists a µ(·) ∈ L1

loc([0, b),R+) such that

H(F (t, U, V ), θ) ≤ µ(t) for a.e. t ∈ [0, b) and for any (U, V ) ∈ B.

Remark 3. If F : [0, b) × C0 × L10 → Kc(E) is a Caratheodory set-valued function,

then using Propositions 2.2.2 and 2.2.3 it is easy to show that a set-valued functionX ∈ AT is a solution of (2.3) on some interval [−r, T ] if and only if

X(t) =

Ψ(t), if −r ≤ t ≤ 0

Ψ(0) +∫ t

0F (s,Xs, DHXs)ds, if 0 ≤ t ≤ T.

(2.4)

Theorem 2.1. Suppose that F : [0, b)×C0×L10 → Kc(E) is a Caratheodory set-valued

function such that the following Lipschitz condition holds: there is a L > 0 such that

H(F (t, U1, V1), F (t, U2, V2)) ≤ L [H0(U1, U2) +H1,0(V1, V2)] (2.5)

for any t ∈ [0, b) and for any (U1, V1), (U2, V2) ∈ C0 × L10. Then, for every Ψ ∈ A0,

there exists a unique solution X : [−r, T ]→ Kc(E) for the initial value problem (2.3)on some interval [−r, T ] with T ∈ (0, b] and T < r.

Proof. Let ρ := H0(Ψ, θ). From (c) it follows that there is a µ(·) ∈ L1loc([0, b),R+)

such that H(F (t, U, V ), θ) ≤ µ(t) for a.e. t ∈ [0, b) and for any (U, V ) ∈ C0 ×L10 with

H0(U, θ) ≤ ρ and H1,0(V, θ) ≤ ρ. We choose T ∈ (0, b/2) such thatT∫0

µ(t)dt < ρ/2.

Let Ψ0 : [−r, b]→ Kc(E) be the set-valued function given by

Ψ0(t) :=

Ψ(t) if t ∈ [−r, 0]Ψ(0) if t ∈ [0, b).

It is easy to see that Ψ0 ∈ AT and HT (Ψ0, θ) ≤ ρ. Further, consider the set Bρ definedby

Bρ :=X ∈ A[T ];X0 = Ψ and H[T ](X,Ψ

0) ≤ ρ.

We remark that if X ∈ Bρ, then H0(Xt, θ) ≤ ρ and H1,0(DHXt, θ) ≤ ρ for anyt ∈ [0, T ]. Further, let us consider the following successive approximation of absolutelycontinuous set-valued functions:

X0(t) = Ψ0(t), t ∈ [0, T ]

29

and

Xn(t) =

Ψ(t), if −r ≤ t ≤ 0

Ψ(0) +∫ t

0F (s,Xn−1

s , DHXn−1s )ds, if 0 ≤ t ≤ T,

for n ≥ 1. We show that Xn ∈ Bρ for any n ≥ 0. Obviously, X0 ∈ Bρ. Let us assumethat X1, X2,...,Xn ∈ Bρ. Then

H(Xn+1(t),Ψ0(t)) ≤∫ t

0

H(F (s,Xns , DHX

n−1s , θ)ds

≤∫ T

0

µ(t)dt < ρ/2,

and soH[T ](X

n+1,Ψ0) = sup−r≤t≤T

H(Xn+1(t),Ψ0(t)) < ρ/2.

Also ∫ T

−rH(DHX

n+1(t), DHΨ0(t))dt =

∫ T

0

H(F (s,Xns , DHX

n−1s ), θ)dt

≤∫ T

0

µ(t)dt < ρ/2.

It follows that

H[T ](Xn+1,Ψ0) = sup

−r≤t≤TH(Xn+1(t),Ψ0(t)) +

∫ T

−rH(DHX

n+1(t), DHΨ0(t))dt

< ρ,

and thus Xn+1 ∈ Bρ. By mathematical induction it follows that Xn ∈ Bρ for any

n ≥ 1. Let us define the sequence Xnn≥1 by

Xn(t) =

Xn(t) if −r ≤ t ≤ TXn(T ) if T ≤ t < b.

Then Xn ∈ Ab, Xn|[−r,0] = Ψ, and H[T ](Xn,Ψ0) ≤ ρ, for any n ≥ 1. By (2.5) and

Lemma 2.2.3, we have

H(Xn+1(t), Xn(t))

≤∫ t

0

H(F (s,Xns , DHX

n−1s ), F (s,Xn−1

s , DHXn−1s ))ds

≤ L

∫ t

0

H0(Xns , X

n−1s )ds = L

∫ t

0

Hs(Xn, Xn−1)ds

30

and ∫ t

t−rH(DHX

n(s), DHXn−1(s))ds

=

∫ t

0

H(F (s,Xns , DHX

n−1s ), F (s,Xn−1

s , DHXn−1s ))ds

≤ L

∫ t

0

H0(Xns , X

n−1s )ds = L

∫ t

0

Hs(Xn, Xn−1)ds.

Therefore,

Ht(Xn+1, Xn) = sup

t−r≤τ≤tH(Xn+1(τ), Xn(τ)) +

∫ t

t−rH(DHX

n+1(τ), DHXn(τ))dτ

≤ 2L

∫ t

0

Hs(Xn, Xn−1)ds.

Let us consider the sequence of real functions gnn≥1, given by gn(t) = Ht(Xn, Xn−1),

n ≥ 1, t ∈ [0, T ]. Then gn+1(t) ≤ 2L∫ t

0gn(s)ds for n ≥ 1 and t ∈ [0, T ]. Since

g1(t) = Ht(X1, X0) ≤ ρ, from the last inequality it follows that gn(t) ≤ ρ (2LT )n

n!,

n ≥ 1, t ∈ [0, T ], and thus

limn→∞

Ht(Xn, Xn−1) = 0 for t ∈ [0, T ].

Further, since

H[T ](Xn+1, Xn) = sup

−r≤τ≤TH(Xn(τ), Xn−1(τ))

+

∫ T

−rH(DHX

n(τ), DHXn−1(τ))dτ = sup

−r≤τ≤TH(Xn(τ), Xn−1(τ))

+

∫ T

−rH(DHX

n(τ), DHXn−1(τ))dτ = sup

T−r≤σ≤2TH(Xn(σ), Xn−1(σ))

+

∫ 2T

T−rH(DHX

n(σ), DHXn−1(σ))dσ = sup

T−r≤σ≤TH(Xn(σ), Xn−1(σ))

+

∫ T

T−rH(DHX

n(σ), DHXn−1(σ))dσ,

it follows that H[T ](Xn+1, Xn) = HT (Xn+1, Xn)→ 0 as n→∞. Therefore, Xnn≥1

is a Cauchy sequence in AT . From Lemma 2.2.1 it follows that there exists a set-valued function X ∈ A[T ] such that X|[−r,0] = Ψ and lim

n→∞H[T ](X

n, X) = 0. Moreover,

31

since H[T ](Xn,Ψ0) < ρ, n ≥ 1, then it is easy to see that H[T ](X,Ψ

0) ≤ H[T ](Xn, X)+

H[T ](Xn,Ψ0) implies H[T ](X,Ψ

0) ≤ ρ, that is, X ∈ Bρ. Next, from (2.5) we have that

H(∫ t

0

F (τ,Xnτ , DHX

nτ )dτ,

∫ t

0

F (τ,Xτ , DHXτ )dτ

)≤

∫ t

0

H (F (τ,Xnτ , DHX

nτ ), F (τ,Xτ , DHXτ )) dτ

≤ L

∫ t

0

H0(Xnτ , Xτ )dτ ≤ L

∫ T

0

Hτ (Xn, X)dτ ≤ L

∫ T

0

H[T ](Xn, X)dτ → 0

as n→∞. Then, we obtain that

limn→∞H(Xn(t),Ψ(0) +

∫ t

0

F (τ,Xτ , DHXτ )dτ

)≤ lim

n→∞H(∫ t

0

F (τ,Xnτ , DHX

nτ )dτ,

∫ t

0

F (τ,Xτ , DHXτ )dτ

)= 0.

It follows that

X(t) =

Ψ(t), if −r ≤ t ≤ 0

Ψ(0) +∫ t

0F (τ,Xτ , DHXτ )dτ, if 0 ≤ t ≤ T,

which represents a solution of (2.3) on [0, T ]. We shall show now that (2.3) has exactlyone solution X ∈ A[T ]. Suppose that X, Y ∈ A[T ] are two solutions of (2.3). Thenwe have that

H(X(t), Y (t)) ≤∫ t

0

H (F (τ,Xτ , (DHX)τ ), F (τ, Yτ , (DHY )τ )) dτ

≤ L

∫ t

0

H0 (Xτ , Yτ ) dτ = L

∫ t

0

Hτ (X, Y ) dτ

and∫ t

t−rH(DHX(τ), DHY (τ))dτ =

∫ t

0

H(DHX(τ), DHY (τ))dτ

≤∫ t

0

H (F (τ,Xτ , DHXτ ), F (τ, Yτ , DHYτ )) dτ

≤ L

∫ t

0

Hτ (X, Y ) dτ.

It follows that

Ht (X, Y ) ≤ 2L

∫ t

0

Hτ (X, Y ) dτ, t ∈ [0, T ],

32

and Gronwall’s lemma implies that Ht (X, Y ) = 0 for t ∈ [0, T ]. By Lemma 2.2.3,we obtain H0 (Xt, Yt) = 0 for t ∈ [0, T ], that is, Xt = Yt for t ∈ [0, T ]. Therefore,X(t) = Xt(0) = Yt(0) = Y (t) for t ∈ [0, T ], and hence (2.3) has a unique solution.This completes the proof.

Theorem 2.2. Suppose that F : [0, b)×C0 ×L10 → Kc(E) satisfies all the conditions

of Theorem 2.1. Then, the largest interval of existence of the solution of (2.3) is [0, b).

Proof. Let X : [−r, β)→ Kc(E) be the solution of (2.3) existing on [−r, β), 0 < β <b. Also, we suppose, by contradiction, that the value of β cannot be increased. Letus consider 0 ≤ s < t < β. Then we have

H (X(t), X(s) ≤∣∣∣∣∣∣∫ t

s

H (F (τ,Xτ , DHXτ ), θ) dτ

∣∣∣∣∣∣≤

∣∣∣∣∣∣∫ t

s

µ(τ)dτ

∣∣∣∣∣∣ .Since µ(·) ∈ L1([0, β],R+) then

∣∣∣∣∫ ts µ(s)ds∣∣∣∣ → 0 as s, t → β−, which implies that

limt→β− X(t) exists. Hence, if we take X(β) = limt→β− X(t), then the function X canbe extended by continuity on [0, β]. Further consider the initial value problem

DHY (t) = G(t, Yt, DHYt), 0 ≤ t < b− β Y|[−(σ+β),0] = Φ

where G(t, Yt, DHYt) = F (t + β, Yt+β, DHYt+β) for 0 ≤ t < b − β and Φ is definedby Φ(s) = X(s + β), for s ∈ [−(σ + β), 0].By Theorem 2.1, there exists a solution

Y : [−(σ + β), β) → E of the initial value problem (2.3), where ˜ ˜β ∈ (0, b − β]. Itfollows that X : [−r, β + β]→ E, given by˜

X(t) =

X(t), for t ∈ [−r, β]

Y (t− β), for t ∈ [β, β + β],

is a solution of the initial value problem (2.3). Therefore, the solution X can becontinued beyond β, contradicting the assumption that β cannot be increased. Thiscontradiction completes the proof.

2.4 Continuous Dependence

For a given Ψ ∈ A0 and a set-valued function F : [0, b) × C0 × L10 → Kc(E) which

satisfies the conditions of Theorem 2.1 let us denote by X(t,Ψ, F, β) the unique

solution on [−r, β] of the initial value prolem (2.3).

33

Theorem 2.3. Suppose that the set-valued functions F,G : [0, b)×C0×L10 → Kc(E)

satisfy the conditions of Theorem 2.1 and that there exists a λ ≥ 0 such that

H0(F (t, U, V ), G(t, U, V )) ≤ λ

for all (t, U, V ) ∈ [0, b)× C0 × L10. Then for any Ψ,Φ ∈ A0 we have

Ht(X, Y ) ≤ 2H0(Ψ,Φ)eLt +2λ

L

(eLt − 1

)for 0 ≤ t ≤ β,

where X(·) = X(·,Ψ, F, β1), Y (·) = X(·,Ψ, G, β2) and β = min(β1, β2).

Proof. For t ∈ [0, β] and by (2.5), we have

H(X(t), Y (t)) ≤ H(Ψ(0),Φ(0)) +

∫ t

0

H(F (s,Xs, DHXs), G(s, Ys, DHYs))ds

≤ H0(Ψ,Φ) +

∫ t

0

H(F (s,Xs, DHXs), F (s, Ys, DHYs))ds

+

∫ t

0

H(F (s, Ys, DHYs), G(s, Ys, DHYs))ds

≤ H0(Ψ,Φ) + L

∫ t

0

H0(Xs, Ys)ds+

∫ t

0

λds

= H0(Ψ,Φ) + λt+ L

∫ t

0

Hs(X, Y )ds.

It follows that

sups∈[0,t]

H(X(s), Y (s)) ≤ H0(Ψ,Φ) + λt+ L

∫ t

0

Hs(X, Y )ds.

Since

sups∈[t−r,0]

H(X(s), Y (s)) = sups∈[t−r,0]

H(Ψ(s),Φ(s))

≤ sups∈[−r,0]

H(Ψ(s),Φ(s) = H0(Ψ,Φ),

we obtain that

sups∈[t−r,t]

H(X(s), Y (s)) ≤ 2H0(Ψ,Φ) + λt+ L

∫ t

0

Hs(X, Y )ds,

that is,

Ht(X, Y ) ≤ 2H0(Ψ,Φ) + λt+ L

∫ t

0

Hs(X, Y )ds. (2.6)

34

Further∫ t

0

H(DHXs, DHYs)ds =

∫ t

0

H(F (s,Xs, DHXs), G(s, Ys, DHYs))ds

≤∫ t

0

H(F (s,Xs, DHXs), F (s, Ys, DHYs))ds

+

∫ t

0

H(F (s, Ys, DHYs), G(s, Ys, DHYs))ds

≤ L

∫ t

0

Hs(X, Y )ds+

∫ t

0

λds

and ∫ 0

t−rH(DHX(s), DHY (s))ds ≤

∫ 0

−rH(DHΨ(s), DHΦ(s))ds

= H1,0(DHΨ, DHΦ).

Thus, we obtain∫ t

t−rH(DHXs, DHYs)ds ≤

∫ 0

t−rH(DHXs, DHYs)ds (2.7)

+

∫ t

0

H(DHXs, DHYs)ds ≤ H1,0(DHΨ, DHΦ) + λt+ L

∫ t

0

Hs(X, Y )ds.

By using (2.6) and (2.7), we obtain

Ht(X, Y ) ≤ 2H0(Ψ,Φ) + 2λt+ 2L

∫ t

0

Hs(X, Y )ds.

Applying Gronwall’s Lemma [36], yields

Ht(X, Y ) ≤ 2H0(Ψ,Φ)e2Lt +λ

L

(e2Lt − 1

)for t ∈ [0, β] and this completes the proof.

Corollary 2.4.1. If F : [0, b)×C0×L10 → Kc(E) satisfies the conditions of Theorem

2.1, then for any Ψ,Φ ∈ A0 we have

Ht(X, Y ) ≤ 2H0(Ψ,Φ)e2Lt for 0 ≤ t ≤ β,

where X(·) = X(·,Ψ, F, β1), Y (·) = X(·,Ψ, F, β2) and β = min(β1, β2).

35

Now consider the functional differential equation with parameterDHX(t) = F (t,Xt, DHXt,Λ)

X|[−r,0] = Ψ,(2.8)

where F : [0, b)× C0 × L10 ×Kc(E1)→ Kc(E) and E1 is a real Banach space. In the

following we suppose that

(H1) For each Λ ∈ Kc(E1) the set-valued function F (·, ·, ·,Λ) : [0, b)× C0 × L10 →

Kc(E) is a Caratheodory set-valued function, and

(H2) there exists a constant K > 0 such that

H(F (t, U1, V1), F (t, U2, V2)) ≤ K [H0(U1, U2) +H1,0(V1, V2) +H(Λ,Ω)]

for any (t, U1, V1), (t, U2, V2) ∈ [0, b]× C0 × L10 and for any Λ,Ω ∈ Kc(E1).

For each Λ ∈ Kc(E1), the existence of a unique solution of (2.8) is assured by

Theorem 2.1.

Theorem 2.4. Suppose that the set-valued function F : [0, b)× C0 × L10 ×Kc(E1)→

Kc(E) satisfies the conditions (H1) and (H2). For Ψ,Φ ∈ A0 we denote by X(·) =X(·,Ψ,Λ) and Y (·) = Y (·,Φ,Ω) the solution of (2.8) corresponding to parameters Λand Ω respectively on [0, β], β < b. Then we have

Ht(X, Y ) ≤ [2H0(Ψ,Φ) + 2KβH(Λ,Ω)] e2Kt for 0 ≤ t ≤ β.

Proof. From Remark 3 we have that

X(t) =

Ψ(t), if −r ≤ t ≤ 0

Ψ(0) +∫ t

0F (τ,Xτ , DHXτ ,Λ)dτ, if 0 ≤ t ≤ β,

and

Y (t) =

Φ(t), if −r ≤ t ≤ 0

Φ(0) +∫ t

0F (τ, Yτ , DHYτ ,Ω)dτ, if 0 ≤ t ≤ β,

Let t ∈ [0, β]. Proceeding exactly in the same way as in Theorem 2.3, we obtain

Ht(X, Y ) ≤ 2H0(Ψ,Φ) +KβH(Λ,Ω) +K

∫ t

0

Hs(X, Y )ds (2.9)

36

and ∫ t

t−rH(DHXs, DHYs)ds ≤ H1,0(DHΨ, DHΦ) +KβH(Λ,Ω) (2.10)

+K

∫ t

0

Hs(X, Y )ds.

Using (2.9) and (2.10), we obtain

Ht(X, Y ) ≤ 2H0(Ψ,Φ) + 2KβH(Λ,Ω) + 2K

∫ t

0

Hs(X, Y )ds.

Applying Gronwall’s Lemma [36], yields

Ht(X, Y ) ≤ [2H0(Ψ,Φ) + 2KβH(Λ,Ω)] e2Kt

for t ∈ [0, β] and this completes the proof.

Chapter 3

Fuzzy delay differential equations

The aim of this chapter is to prove the existence and uniqueness theorem for fuzzy

delay differential equation driven by Liu process. We also establish continuous de-

pendence of solution with respect to initial data.

3.1 Introduction

Credibility theory was founded by Liu (2004) and refined by Liu (2007) as a branch

of mathematics for studying the behavior of fuzzy phenomena. The multidimensional

Liu process are defined and studied in You (2007). Also, Liu (2007) founded an

uncertainty theory that is a branch of mathematics based on normality, monotonicity,

self-duality, and countable subadditivity axioms. In the paper (Chen and Ralescu

2009), a formula for computing the truth value of independent uncertain propositions

is proved. A hybrid variable was introduced by Liu (2006) as a tool to describe

the quantities with fuzziness and randomness. Fuzzy random variable and random

fuzzy variable are instances of hybrid variable. In order to measure hybrid events,

a concept of chance measure was introduced by Li and Liu (2009) (see also Liu

2008). The reflection principle of Liu process is proved in Dai (2007). Based on

37

38

the random fuzzy theory, a renewal process with random fuzzy interarrival times is

proposed in the papers (Li et al. 2009) and (Zhao and Liu 2003). Fuzzy differential

equation was proposed by Li and Liu (2006) (see also, Chen 2008; You 2008) as a

type of differential equation driven by Liu process just like that stochastic differential

equations driven by Brownian motion. The existence and uniqueness theorem for

homogeneous fuzzy differential equation was proved in Liu (2007) (see also, Chen

2008). In the paper (Chen and Liu 2010) the authors prove existence and uniqueness

of solution for uncertain differential equations (see also, Liu 2007). Some concepts of

stability for fuzzy differential equations are given in the paper (Zhu 2010).

3.2 Preliminaries

Let Ω be a nonempty set, and P the family of all subsets of Ω. Each element A in P

is called an event.

A mapping Cr : P → [0, 1] is called credibility measure if it satisfies the following

axioms [99]:

A 1 (Normality) Cr (Ω) = 1

A 2 (Monotonicity) Cr (A) ≤ Cr (B) if A ⊂ B

A 3 (Self-Duality) Cr (A)+Cr (Ac) = 1 for any eventA.HereAc = ω ∈ Ω;ω /∈ A

A 4 (Maximality) Cr

(⋃i∈I

Ai

)= sup

i∈ICr (Ai) for any events Ai, i ∈ I, with

supi∈I

Cr (Ai) < 0, 5.

The triplet (Ω,P , Cr) is called a credibility space.

39

A fuzzy variable is defined as a (measurable) function ξ : (Ω,P , Cr)→ R. We de-

note by F(Ω) the space of fuzzy variables. If ξ is a fuzzy variable then its membership

function is derived from the credibility measure by

µ(x) = max 1, 2Cr (ξ = x) , x ∈ R

Let T ⊂ R be an index set and (Ω,P , Cr) be a credibility space. A fuzzy process

[96] is a function X : T × Ω → R. Note that for each t ∈ T fixed we have a fuzzy

variable ω 7→ X(t, ω) : (Ω,P , Cr)→ R. On other hand, fixing ω → Ω we can consider

the function t 7→ X(t, ω) wich is caled a path of X. A fuzzy process X is said to be

continuous if the function t 7→ X(t, ω) is continuous for all ω ∈ Ω. In the followig, we

use the notation X(t) instead of X(t, ω).

Let ξ be a fuzzy variable. Then the expected value of ξ is defined by [99]

E [ξ] =

+∞∫0

Cr (ξ ≥ r) dr −0∫

−∞

Cr (ξ ≤ r) dr

provided that at least one of the two integrals is finite. The variance of ξ is defined

by

V [ξ] = E[(ξ − E [ξ])2]

A fuzzy process X is said to have independent increments if

X (t1)−X (t0) , X (t2)−X (t1) , ..., X (tk)−X (tk−1)

are independent fuzzy variables for any times t0 < t1 < ... < tk, that is,

Cr

(k⋂i=1

X (ti) ∈ Bi

)= min

1≤i≤kCr (X (ti) ∈ Bi)

for any sets B1, B2, ..., Bk of real numbers.

40

A fuzzy process X is said to have stationary increments if, for any given t > 0,

the X (s+ t)−X (t) are identically distributed fuzzy variables for all s > 0.

A fuzzy process C is said to be a Liu process if [96]

(L1) C (0) = 0,

(L2) C (t) has stationary and independent increments,

(L3) every increment C (s+ t)− C (s) is a normally distributed fuzzy variable with

expected value et and variance σ2t2 whose membership functions is

µ (x) = 2

(1 + exp

(π |x− et|√

6σt

))−1

, −∞ < x < +∞

The parameters e and σ are called the drift and diffusion coefficients, respectively.

Liu process is said to be standard if e = 0 and σ = 1.

Let C be a standard Liu process, and dt an infinitesimal time interval. Then

dC(t) = C(t + dt) − C(t) is a fuzzy process such that, for each t, the dC(t) is a

normally distributed fuzzy variable with E[dC(t)] = 0.

Let X be a fuzzy process and let dC(t) be a standard Liu process. For any

partition of closed interval [a, b] with a = t1 < t2 <···< tk+1 = b, the mesh size is

written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the Liu integral [96] of X with respect to C(t) is

b∫a

X(t)dC(t) = lim∆→0

k∑i=1

X(ti)[C(ti+1)− C(ti)]

provided that the limit exists almost surely and is a fuzzy variable.

41

Let X(t) be a fuzzy process and let C(t) be a standard Liu process. If Liu integralb∫a

X(t)dC(t) exists and it is a fuzzy variable, then X(t) is called Liu integrable. We

know that any continuous fuzzy process is Liu integrable [147].

Also, we remark that a Liu process C is Lipschitz-continuous [39], that is, for

every given ω ∈ Ω, there exists K(ω) > 0 such that

|C(t, ω)− C(s, ω)| ≤ K(ω)|t− s|, for all t, s ≥ 0. (3.1)

Using the same idea that in [35], we can obtain the following result for Liu inte-

grable process.

Lemma 3.2.1. Suppose that C(t) is a standard Liu process and X(t) is a fuzzy processon [a, b] with respect to t. If K(ω) > 0 is the Lipschitz constant for path t 7→ C(t)(ω)with ω ∈ Ω fixed, then we have∣∣∣∣∣∣

b∫a

X(t)dC(t)

∣∣∣∣∣∣ ≤ K(ω)

b∫a

|X(t)| dt. (3.2)

Proof. Let a = t1 < t2 <···< tk+1 = b be a partition of [a, b] and ∆ = max1≤i≤k

|ti+1 − ti|.Then, by (3.1), we have∣∣∣∣∣∣

b∫a

X(t)dC(t)

∣∣∣∣∣∣ =

∣∣∣∣∣ lim∆→0

k∑i=1

X(ti)[C(ti+1)− C(ti)]

∣∣∣∣∣≤ K(ω) lim

∆→0

k∑i=1

|X(ti)| · |ti+1 − ti|

≤ K(ω)

b∫a

|X(t)| dt.

42

3.3 Fuzzy delay differential equation

For a positive number r, we denote by Cr the space C ([−r, 0] ,R). Then Cr is a

Banach space with the respect to the supremum norm: ‖ϕ‖ = supt∈[−r,0]

|ϕ (t)|. Let

X : [−r,∞) × Ω → R be a fuzzy process. For each t ≥ 0 we can define a fuzzy

segment process Xt : [−r, 0]× Ω→ R given by

Xt (s, ω) = X (t+ s, ω) , for all s ∈ [−r, 0] and ω ∈ Ω.

Xt is called a fuzzy process with delay (or memory) of the fuzzy process X at moment

t ≥ 0. In the followig, we use the notation Xt(s) instead of Xt(s, ω).

Lemma 3.3.1. If X is a continuous fuzzy process, then t 7→ Xt : [0,∞)→ Cr is alsocontinuous.

Proof. Let us fixed τ ∈ [0,∞) and ε > 0. Since X is continuous, there exists δ > 0such that, for every t ∈ [0,∞) with |t − τ | < δ, we have that |X(t)−X(τ)| < ε.Since X is continuous, then it is uniformly continuous on the compact interval I =[max−r, τ − r − δ, τ + δ]. Hence, there exists η > 0 such that, for every t1, t2 ∈ Iwith |t1 − t2| < η, we have that |X(t1)−X(t2)| < ε. Then, for every s ∈ [−r, 0], wehave that τ + s ∈ I and t+ s ∈ I if |t− τ | < δ. Since |(t+ s)− (τ + s)| < η, it followsthat

‖Xt −Xτ‖ = sup−r≤s≤0

|Xt(s)−Xτ (s)|

= sup−r≤s≤0

|X(t+ s)−X(τ + s)| ≤ ε,

and so, t 7→ Xt is continuous.

Corollary 3.3.1. If F : [0,∞)×Cr → R is a jointly continuous function and X is acontinuous fuzzy process, then t 7→ F (t,Xt) is also continuous.

Remark 4. If F : [0,∞)×Cr → R is a jointly continuous function and X : [−r,∞)→R is a continuous fuzzy process, then t 7→ F (t,Xt) is integrable on each compactinterval [τ, T ]. Also, if F : [0,∞) × Cr → R is a jointly continuous function andX : [−r,∞) → R is a continuous fuzzy process, then the function t 7→ F (t,Xt) :[0,∞) → R is bounded on each compact interval [0, T ]. Also, the function t 7→F (t, 0) : [0,∞)→ R is bounded on each compact interval [0, T ].

43

We say that the function F : [0,∞) × Cr → R is locally Lipschitz if for all

a, b ∈ [0,∞) and ρ > 0, there exists L > 0 such that

|F (t, ϕ)− F (t, ψ)| ≤ L ‖ϕ− ψ‖ , a ≤ t ≤ b, ϕ, ψ ∈ Bρ.

where Bρ := ϕ ∈ Cr; ||ϕ|| ≤ ρ.

Lemma 3.3.2. Assume that F : [0,∞)×Cr → R is continuous and locally Lipschitz.Then, for each compact interval J ⊂ [0,∞) and ρ > 0, there exists M > 0 such that

|F (t, ϕ)| ≤M , t ∈ J , ϕ ∈ Bρ.

Proof. Indeed, for t ∈ J , we have

|F (t, ϕ)| ≤ |F (t, ϕ)− F (t, 0)|+ |F (t, 0)|≤ L ‖ϕ‖+ |F (t, 0)| ≤ ρL+ η,

where η := supt∈J |F (t, 0)|.

Suppose that C is a standard Liu process and F,G : [0,∞) × Cr → R are some

given function.

We consider the following fuzzy delay differential equationdX (t) = F (t,Xt) dt+G (t,Xt) dC (t) , t ≥ τ

X (t) = ϕ (t− τ) , τ − r ≤ t ≤ τ(3.3)

By solution of fuzzy delay differential equation (3.3) on some interval [τ, b), we we

mean a continuous fuzzy process X : [τ − r, b)×Ω→ R, such that X (t) = ϕ (t− τ) ,

for τ−r ≤ t ≤ τ and dX (t) = F (t,Xt) dt+G (t,Xt) dC (t) , for all t ≥ τ . We remark

that if F,G : [0,∞)×Cr → R are continuous, then X : [τ−r, b)×Ω→ R is a solution

for (3.3) if and only if

X(t) =

ϕ (t− τ) , for τ − r ≤ t ≤ τ

ϕ (0) +t∫τ

F (s,Xs) ds+t∫τ

G (s,Xs) dC (s) , for τ ≤ t < b

To solve a fuzzy differential equations with dealy, we use the method of steps [56].

44

Example 3.3.1. Consider the following fuzzy differential equation:dX (t) = µX(t− τ)dt+ σdC (t) , t ≥ 0X (t) = 1, − τ ≤ t ≤ 0,

(3.4)

where µ and σ are constants, and C (t) is a standard Liu process.

If 0 ≤ t ≤ τ , then −τ ≤ t− τ ≤ 0, and so X(t− τ) = 1. Therefore, for 0 ≤ t ≤ τ ,

we have to solve the following differential equationdX (t) = µdt+ σdC (t)

X(0) = 1.

We obtain

X(t) = X(0) +

t∫0

µdt+ σC (t) ,

and so

X(t) = 1 + µt+ σC(t) for 0 ≤ t ≤ τ .

If τ ≤ t ≤ 2τ , then 0 ≤ t − τ ≤ τ , and so X(t − τ) = 1 + µ(t − τ) + σC (t− τ).

Therefore, for τ ≤ t ≤ 2τ , we have to solve the following differential equation

dX (t) = µ[1 + µ(t− τ) + σC (t− τ)]dt+ σdC (t)

X(τ) = 1 + µτ + σC (τ).

Then

X(t) = X(τ) +

t∫τ

µ[1 + µ(s− τ) + σC (s− τ)]ds+ σC (t)

and so

X(t) = 1 + µτ + µ(t− τ) + µ2 (t− τ)2

2+ σC (t) + µσ

t∫τ

C (s− τ) ds

for τ ≤ t ≤ 2τ . Clearly, we can continue this method, finding the expression for X(t)

on each interval [nτ, (n+ 1)τ ] with n ≥ 0.

45

Example 3.3.2. (Ehrlich ascities tumor model).In Schuster and Schuster (1995), the following form of delay logistic equation was

proposed to describe the Ehrlich ascities tumor:

X ′(t) = αX(t− r)(

1− X(t− r)K

)where α is the net reproduction of the rate tumor and K is the caring capacity. Here,r reflects the duration of the cell cycle. In generally, the deterministic growth modelsdo not necessarily give a satisfactory deterministic prediction of mean trends. Fluctu-ations of the growth dynamics for a cell tumor population can be influenced by manyindependent characteristics of the state variables, such as: medical treatment, mentalstatus, diet, physical activity, age, etc.. The specific value of these characteristics notalways can be evaluated or measured in classical sense, which are uncertain and wecan only conjecture intuitively. Therefore, we consider that a more realistic growthmodel for the Ehrlich ascities tumor should be the following fuzzy model:

dX(t) = αX(t− r)(

1− X(t− r)K

)dt+ σdC(t), (3.5)

where σ is a constant controlling the amplitude of noise, and C(t) is the standard Liuprocess (white noise). We can associate with the above fuzzy differential equations theinitial condition

X(t) = ϕ(t), −r ≤ t ≤ 0 (3.6)

The initial value problem 3.53.6 can be solved using the method of steps (Hale 1997).Given the importance of such issues in the modeling of growth dynamics for a celltumor population, a detailed study of it will be done in a next paper.

Theorem 3.1. Suppose that F,G : [0,∞) × Cr → R are continuous and satisfy thefollowing locally Lipschitz continuous: for all a, b ∈ [0,∞) and ρ > 0 there existsL > 0 such that

|F (t, ϕ)− F (t, ψ)|+ |G (t, ϕ)−G (t, ψ)| ≤ L ‖ϕ− ψ‖ , a ≤ t ≤ b, ϕ, ψ ∈ Bρ

(3.7)where Bρ = ϕ ∈ Cr; ‖ϕ‖ ≤ ρ .Then, for each (τ, ϕ) ∈ [0,∞) × Cr and ω ∈ Ω, thereexists T = T (ω) > τ such that the fuzzy delay differential equation (3.3) has a uniquesolution on [τ − r, T ] .

Proof. Let ρ > 0 be any positive number. Since F and G satisfy the locally Lipschitzcondition (3.7), then there exists L > 0 such that

|F (t, ϕ)− F (t, ψ)|+|G (t, ϕ)−G (t, ψ)| ≤ L ‖ϕ− ψ‖ , τ ≤ t ≤ h, ϕ, ψ ∈ Bρ (3.8)

46

for some h > τ. Also, by Lemma 3.2.1, there exists M > 0 such that

max|F (t, ϕ) |, G (t, ϕ) | ≤M for (t, ϕ) ∈ [τ, h]×B2ρ.

It is known that K(ω) > 0 is bounded on Ω (see Dai 2009, Theorem 2). We chooseT = T (ω) := τ + min

h, ρ

MK

, where K > 0 is a constant such that 1 +K(ω) < K

for all ω ∈ Ω. To solve (3) we construct a solution process via approximation byPicard iterations. For this, we defined a sequence of functions Xm : [τ − r, T ] → Estarting with the initial continuous function

X0(t) =

ϕ (t− τ) , for τ − r ≤ t ≤ τϕ (0) , for τ ≤ t ≤ T

and we define

Xm+1(t) =

ϕ (t− τ) , for τ − r ≤ t ≤ τ

ϕ (0) +t∫τ

F (s,Xms ) ds+

t∫τ

G (s,Xms ) dC (s) , for τ ≤ t ≤ T

(3.9)

if m = 0, 1, ... .Clearly, |X0(t)| ≤ 2ρ on [τ, T ] . Suppose that |Xm(t)| ≤ 2ρ on [τ, T ].Then by Lemma 3.2.1, we have

|Xm+1(t)| ≤ |X0(t)|+∣∣∣∣ t∫τ

F (s,Xms ) ds

∣∣∣∣+

∣∣∣∣ t∫τ

G (s,Xms ) dC(s)

∣∣∣∣≤ ρ+

t∫τ

|F (s,Xms ) |ds+K(ω)

t∫τ

|G (s,Xms ) |ds ≤ ρ+M [1 +K(ω)](t− τ)

≤ ρ+MK(t− τ) < 2ρ,

and so, |Xm+1(t)| ≤ 2ρ on [τ, T ]. Therefore, Xm(t) ∈ B2ρ for all t ∈ [τ, T ] and m ≥ 0.

47

By (3.8) and (3.9), we find that

|Xm+1(t)−Xm(t)| ≤∣∣∣∣ t∫τ

[F (s,Xms ) ds− F (s,Xm−1

s )]ds

∣∣∣∣+

∣∣∣∣ t∫τ

[G (s,Xms ) dC(s)−G (s,Xm−1

s )]dC(s)

∣∣∣∣≤

t∫τ

|F (s,Xms )− F (s,Xm−1

s )| ds+K(ω)t∫τ

|G (s,Xms )−G (s,Xm−1

s )| ds

≤ L(1 +K(ω))t∫τ

‖Xms −Xm−1

s ‖ ds

≤ LKt∫τ

supθ∈[−r,0] |Xms (θ)−Xm−1

s (θ)|ds

= LKt∫τ

supθ∈[−r,0] |Xm(θ + s)−Xm−1(θ + s)|ds

= LKt∫τ

supζ∈[s−r,s] |Xm(ζ)−Xm−1(ζ)|ds, t ∈ [τ, T ].

In particular,

|X1(t)−X0(t)| ≤∣∣∣∣ t∫τ

F (s,X0s ) ds

∣∣∣∣+

∣∣∣∣ t∫τ

G (s,X0s ) dC(s)

∣∣∣∣≤

t∫τ

|F (s,X0s ) |ds+K(ω)

t∫τ

|F (s,X0s ) |ds ≤MK(t− τ),

and so,

|X2(t)−X1(t)| ≤ L(1 +K(ω))t∫τ

supζ∈[s−r,s] |X1(ζ)−X0(ζ)|ds

= LMK2t∫τ

(s− τ)ds = MLL2K2

2(t− τ)2, t ∈ [τ, T ] .

Further, if we assume that

|Xm(t)−Xm−1(t)| ≤ M

L

LmKm

m!(t− τ)m , t ∈ [τ, T ] (3.10)

48

then we have

|Xm+1(t)−Xm(t)| ≤ L(1 +K(ω))t∫τ

supζ∈[s−r,s] |Xm(ζ)−Xm−1(ζ)|ds

= L(1 +K(ω))t∫τ

KLLmKm

m!(t− τ)m ds

= MLLm+1Km+1

(m+1)!(t− τ)m+1 , t ∈ [τ, T ] .

If follows by mathematical induction that (3.10) holds for any m ≥ 1.Consequently,

the series∞∑m=1

|Xm(t) − Xm−1(t)| is uniformly convergent on [τ, T ] , and so is the

sequence Xmm≥0 . It follows that there exists X : [τ, T ] → R such that |Xm(t) −X(t)| → 0 as m→∞. Since

|F (s,Xms )− F (s,Xs)|+ |G (s,Xm

s )−G (s,Xs)|

≤ L ‖Xms −Xs‖ ≤ L sup

τ≤t≤T|Xm(t)−X(t)|,

we deduce that

|F (s,Xms )− F (s,Xs)| → 0 and |G (s,Xm

s )−G (s,Xs)| → 0

uniformly on [τ, T ] as m→∞. Therefore, since∣∣∣∣ t∫τ

F (s,Xms ) ds−

t∫τ

F (s,Xs) ds

∣∣∣∣ ≤ t∫τ

|F (s,Xms )− F (s,Xs)| ds

it follows that limm→∞

t∫τ

F (s,Xms ) ds =

t∫τ

F (s,Xs) ds, t ∈ [τ, T ]. Also, by Lemma 3.2.1,

we obtain∣∣∣∣ t∫τ

G (s,Xms ) dC(s)−

t∫τ

G (s,Xs) dC(s)

∣∣∣∣ ≤ K(ω)t∫τ

|G (s,Xms )−G (s,Xs)| ds,

and so, limm→∞

t∫τ

G (s,Xms ) dC(s) =

t∫τ

G (s,Xs) dC(s), t ∈ [τ, T ]. Extending X to

[τ − r, τ ] in the usual way by X(t) = ϕ (t− τ) for t ∈ [τ − r, τ ] , then by (3.9) weobtain that

X(t) =

ϕ (t− τ) , for τ − r ≤ t ≤ τ

ϕ (0) +t∫τ

F (s,Xs) ds+t∫τ

G (s,Xs) dC(s), for τ ≤ t ≤ T.

49

Next, we prove that X is continuous. Obviously, X is sample continuous on [τ − r, τ ].For t > s > τ , we have

|X(t)−X(s)| =∣∣∣∣ t∫s

F (ξ,Xξ) dξ +t∫s

G (ξ,Xξ) dC(ξ)

∣∣∣∣≤

t∫s

|F (ξ,Xξ)| dξ +K(ω)t∫s

|G (ξ,Xξ)| dξ

≤M [1 +K(ω)](t− s) ≤MK(t− s)

and so, |X(t) − X(s)| → 0 as s → t. Hence X is continuous. Therefore, X(t) is asolution for (3.3). To prove the uniqueness, assume that X and Y are solution of(3.3). Then for every t ∈ [τ, T ] we have

|X(t)− Y (t)|

=

∣∣∣∣ t∫τ

[F (s,Xs) ds− F (s, Ys)]ds+t∫τ

[G (s,Xs) ds−G (s, Ys)]dC(s)

∣∣∣∣≤

t∫τ

|F (s,Xs)− F (s, Ys) |ds+K(ω)t∫τ

|G (s,Xs)−G (s, Ys)| ds

≤ L(1 +K(ω))t∫τ

L ‖Xs − Ys‖ ds ≤ LKt∫τ

supζ∈[s−r,s]

|X(ζ)− Y (ζ)| ds

If we let ξ(s) := supζ∈[s−r,s]

|X(ζ)− Y (ζ)| , s ∈ [τ, T ], by Gronwall’s lemma we obtain

that X(t) = Y (t) on [τ, T ]. This proves the uniqueness of the solution of (3.3).

Theorem 3.2. Assume that the functions F,G : [0,∞) × Cr → R are continuousand satisfiy the locally Lipschitz (3.7). If (τ, ϕ), (τ, ψ) ∈ [0,∞) × Cr and X(ϕ) :[τ − r, T1) → R and X(ψ) : [τ − r, T2) → R are unique solutions of (3.3) withX(t) = ϕ(t− τ) and Y (t) = ψ(t− τ) on [τ − r, τ ]. Then

|X(ϕ)(t)−X(ψ)(t)| ≤ ‖ϕ− ψ‖ eLK(t−τ) for all t ∈ [τ, T ), (3.11)

where T = minT1, T2.

Proof. On [τ, T ) solution X(ϕ) satisfies the relation

X(t) =

ϕ(t− τ), if t ∈ [τ − r, τ ]

ϕ(0) +∫ tτF (s,Xs(ϕ)ds+

t∫τ

G (s,Xs) dC(s), if t ∈ [τ, T )

50

and solution X(ψ) satisfies the same relation but with ψ in place of ϕ. Then, fort ∈ [τ, T ), we have

|X(ϕ)(t)−X(ψ)(t)| ≤ |ϕ(0)− ψ(0)|+∫ tτ|F (s,Xs(ϕ)− F (s,Xs(ψ)|ds

+K(ω)∫ tτ|G(s,Xs(ϕ)−G(s,Xs(ψ)|ds

≤ ‖ϕ− ψ‖+ L[1 +K(ω)]∫ tτ‖Xs(ϕ)−Xs(ψ)‖ ds

≤ ‖ϕ− ψ‖+ LK∫ tτ

maxθ∈[τ−r,s] |X(ϕ)(θ)−X(ψ)(θ)|ds.

If we let w(t) = supθ∈[τ−r,s] |X(ϕ)(θ)−X(ψ)(θ)|, τ ≤ s ≤ t, then we have

w(t) ≤ ‖ϕ− ψ‖+ LK

∫ t

τ

w(s)ds, τ ≤ t < T

and Gronwall’s inequality gives

w(t) ≤ ‖ϕ− ψ‖ eL[1+K(ω)](t−τ), τ ≤ t < T

implying that (3.11) holds.

Chapter 4

Uncertain dynamic systems ontime scales

4.1 Introduction

The uncertain theory was introduced by Liu [94] as an important tool for the study

of some real world phenomena who cannot be modeled by fuzziness. For a discussion

about the relations and differences among uncertainty, fuzziness and probability see

the paper [92, 53]. The main stages on the development of this theory and some

fundamental results can be found in works [53, 94, 97, 98, 148]. The concept of

uncertain differential equation was also introduced in [94], but the existence and

uniqueness of the solution of this kind of differential equations was obtained by Chen

and Liu in the paper [34]. The theory of dynamic systems on time scales allows us to

study both continuous and discrete dynamic systems simultaneously. Since Hilger’s

initial work [60] there has been significant growth in the theory of dynamic systems

on time scales, covering a variety of different qualitative aspects. We refer to the

books [31, 32], and the papers [5, 6, 141].

The purpose of this chapter is to prove the existence and uniqueness of solution for

the dynamic systems on time scales with uncertain parameters. The organization of

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this chapter is as follows. Section 4.2 presents a few definitions and concepts of time

scales. Also, the notion of uncertain process on a time scale is introduced. In Section

4.3 we prove the existence and uniqueness of solution for the dynamic systems on time

scales with uncertain parameters. In the last section, we study the linear dynamic

systems on time scales with uncertain parameters.

4.2 Preliminaries

Time scale. By a time scale T we mean any closed subset of R. Then T is a

complete metric space with the metric defined by d(t, s) := |t− s| for t, s ∈ T. Since

we know that for working into the different connected components of the time scale

T, we need the concept of jump operators. The forward jump operator σ : T→ T is

defined by σ(t) := infs ∈ T : s > t, while the backward jump operator ρ : T → T

is defined by ρ(t) := sups ∈ T : s < t. In this definition we put inf ∅ = supT and

sup ∅ = inf T. The graininess function µ : T → [0,∞) is defined by µ(t) := σ(t) − t.

If σ(t) > t, we say t is a right-scattered point, while if ρ(t) < t, we say t is a left-

scattered point. Points that are right-scattered and left-scattered at the same time

will be called isolated points. A point t ∈ T such that t < supT and σ(t) = t, is

called a right-dense point. A point t ∈ T such that t > inf T and ρ(t) = t, is called a

left-dense point. Points that are right-dense and left-dense at the same time will be

called dense points. The set Tκ is defined to be Tκ = Trm if T has a left-scattered

maximum m, otherwise Tκ = T. To understand the notions we have to consider some

examples for clearing the abstraction of the situation. Given a time scale interval

[a, b]T := t ∈ T : a ≤ t ≤ b, then [a, b]κT denoted the interval [a, b]T if a < ρ(b) = b

and denote the interval [a, b)T if a < ρ(b) < b. In fact, [a, b)T = [a, ρ(b)]T. Also, for

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a ∈ T, we define [a,∞)T = [a,∞) ∩ T. If T is a bounded time scale, then T can be

identified with [inf T, supT]T.

If t0 ∈ T and δ > 0, then we define the following neighborhood of t0: UT(t0, δ) :=

(t0 − δ, t0 + δ) ∩ T, U+T (t0, δ) := [t0, t0 + δ) ∩ T, and U−T (t0, δ) := (t0 − δ, t0] ∩ T.

Let Rm be the space of m-dimensional column vectors x = col(x1, x2, ...xm) with

a norm || · ||.

Definition 4.2.1. ([31]). A function f : T→ Rm is called regulated if its right-sidedlimits exist (finite) at all right-dense points in T, and its left-sided limits exist (finite)at all left-dense points in T. A function f : T → Rm is called rd-continuous if it iscontinuous at all right-dense points in T and its left-sided limits exist (finite) at allleft-dense points in T. Denote by Crd(T,Rm) the set of all rd-continuous functionfrom T into Rm.

Obviously, a continuous function is rd-continuous, and a rd-continuous function

is regulated ([31, Theorem 1.60]).

Definition 4.2.2. A function f : [a, b]T × Rm → Rm is called Hilger continuous if fis continuous at each point (t, x) where t is right-dense, and the limits

lim(s,y)→(t−,x)

f(s, y) and limy→x

f(t, y)

both exist and are finite at each point (t, x) where t is left-dense.

Definition 4.2.3. ([31]). Let f : T→ Rm and t ∈ Tκ. Let f∆(t) ∈ Rm (provided itexists) with the property that for every ε > 0, there exists δ > 0 such that∥∥f(σ(t))− f(s)− f∆(t)[σ(t)− s]

∥∥ ≤ ε |σ(t)− s| (4.1)

for all s ∈ UT(t, δ). We call f∆(t) the delta ( or Hilger) derivative (∆-derivative forshort) of f at t. Moreover, we say that f is delta differentiable (∆-differentiable forshort) on Tκ provided f(t) exists for all t ∈ Tκ.

The following result will be very useful.

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Proposition 4.2.1. ([31, Theorem 1.16]). Assume that f : T→ Rm and t ∈ Tκ.(i) If f is ∆-differentiable at t then f is continuous at t.(ii) If f is continuous at t and t is right-scattered then f is ∆-differentiable at t

with

f∆(t) =f(σ(t))− f(t)

σ(t)− t.

(iii) If f is ∆-differentiable at t and t is right-dense then

f∆(t) = lims→t

f(t)− f(s)

t− s.

(iv) If f is ∆-differentiable at t then f(σ(t)) = f(t) + µ(t)f∆(t).

It is known [54] that for every δ > 0 there exists at least one partition P : a =

t0 < t1 < ... < tn = b of [a, b)T such that for each i ∈ 1, 2, ..., n either ti − ti−1 ≤ δ

or ti − ti−1 > δ and ρ(ti) = ti−1. For given δ > 0 we denote by P([a, b)T, δ) the set of

all partitions P : a = t0 < t1 < ... < tn = b that possess the above property.

Let f : T→ Rm be a bounded function on [a, b)T, and let P : a = t0 < t1 < ... <

tn = b be a partition of [a, b)T. In each interval [ti−1, ti)T,where 1 ≤ i ≤ n, we choose

an arbitrary point ξi and form the sum

S =n∑i=1

(ti − ti−1)f(ξi).

We call S a Riemann ∆-sum of f corresponding to the partition P .

Definition 4.2.4. ([55]). We say that f is Riemann ∆-integrable from a to b (oron [a, b)T) if there exists a vector I ∈ Rm with the following property: for eachε > 0 there exists δ > 0 such that ‖S − I‖ < ε for every Riemann ∆-sum S of fcorresponding to a partition P ∈ P([a, b)T, δ) independent of the way in which wechoose ξi ∈ [ti−1, ti)T, i = 1, 2, ..., n. It is easily seen that such a vector I is unique.The vector I ∈ Rm is the Riemann ∆-integral of f from a to b, and we will denote itby∫ baf(t)∆t.

Proposition 4.2.2. ([55, Theorem 5.8]). A bounded function f : [a, b)T → Rm isRiemann ∆-integrable on [a, b)T if and only if the set of all right-dense points of [a, b)Tat which f is discontinuous is a set of ∆−measure zero.

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Since every regulated function on a compact interval is bounded (see [31, Theorem

1.65]). Then we get that every regulated function f : [a, b]T → Rm, is Riemann ∆-

integrable from a to b.

Proposition 4.2.3. ([61, Theorem 5.8]). Assume that a, b ∈ T, a < b and f : T →Rm is rd-continuous. Then the integral has the following properties.(i) If T = R, then

∫ baf(t)∆t =

∫ baf(t)dt, where the integral on the right-hand side is

the Riemann integral.(ii) If T consists of isolated points, then∫ b

a

f(t)∆t =∑

t∈[a,b)T

µ(t)f(t).

Definition 4.2.5. ([31]). A function g : T → Rm is called a ∆-antiderivative off : T→ Rm if g∆(t) = f(t) for all t ∈ Tκ.

One can show that each rd-continuous function has a ∆-antiderivative [31, Theo-

rem 1.74].

Proposition 4.2.4. ([55, Theorem 4.1]). Let f : T→ Rm be Riemann ∆-integrable

function on [a, b)T. If f has a ∆-antiderivative g : [a, b]T → Rm, then∫ baf(t)∆t =

g(b)−g(a). In particular,∫ σ(t)

tf(s)∆s = µ(t)f(t) for all t ∈ [a, b)T (see [31, Theorem

1.75])

Proposition 4.2.5. ([55, Theorem 4.3]). Let f : T → Rm be a function which isRiemann ∆-integrable from a to b. For t ∈ [a, b]T, let g(t) =

∫ taf(t)∆t. Then g is

continuous on [a, b]T. Further, let t0 ∈ [a, b)T and let f be arbitrary at t0 if t0 is right-scattered, and let f be continuous at t0 if t0 is right-dense. Then g is ∆-differentiableat t0 and g∆(t0) = f(t0).

Lemma 4.2.1. ([141]). Let g : R→ R be a continuous and nondecreasing function.If s, t ∈ T with s ≤ t, then ∫ t

s

g(τ)∆τ ≤∫ t

s

g(τ)dτ.

Other properties of the Riemann ∆-integral can be find in [5] and [31].

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4.2.1 Uncertain process on time scales:

Let Γ be an nonempty set and let L be a σ-algebra of sets of Γ. A mappingM : L →

[0, 1] is called uncertain measure if it satisfies the following axioms:

(A1) M(Γ) = 1;

(A2) M(A) ≤M(B) for all A,B ∈ L with A ⊂ B;

(A3) M(A) +M(Ac) = 1 for all A ∈ L, where Ac := Γr A;

(A4) For every countable sequence An of elements of L, we have

M

(∞⋃n=1

An

)=∞∑n=1

M (An) .

Let Γ be a nonempty set, L a σ-algebra on Γ, and M an uncertain measure. T

he triplet (Γ,L,M) is called an uncertain space.

Example 4.2.1. Let us consider Γ = (0, 1), L the σ-algebra of all Borel subsets ofΓ. Let λ : (0, 1) → R+ be defined by λ(x) =

∣∣x− 12

∣∣, x ∈ (0, 1). Then the mappingM : L → [0, 1] defined by

M(A) =

supx∈A

λ(x), if supx∈A

λ(x) < 1/2

1− supx∈Ac

λ(x), if supx∈A

λ(x) ≥ 1/2,

is an uncertain measure on (0, 1).

Denote by B the σ-algebra of all Borel subsets of Rm. A function X(·) : Γ→ Rm

is called an uncertain variable if X is a measurable function from (Γ,F) into (Rm,B);

that is, X−1(B) := γ ∈ Γ;X(γ) ∈ B ∈ L for all B ∈ B. A time scale uncertain

process is a function X(·, ·) : [a, b]T × Γ → Rm such that X(t, ·) : Γ → Rm is an

uncertain vector for each t ∈ T. For each point γ ∈ Γ, the function on T given by

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t 7→ X(t, γ) is will be called a sample path of the time scale uncertain process X(·, ·)

corresponding to γ. A time scale uncertain process X(·, ·) is said to be regulated (rd-

continuous, continuous) if the trajectory t 7→ X(t, γ) is a regulated (rd-continuous,

continuous) function on [a, b]T for each γ ∈ Γ.

Lemma 4.2.2. Let X(·, ·) : [a, b]T × Γ → Rm be a time scale uncertain process. Ifthe sample path t 7→ X(t, γ) is Riemann ∆-integrable on [a, b)T for every γ ∈ Γ, thenthe function Y (·, ·) : [a, b]T × Γ→ Rm given by

Y (t, γ) =

∫ t

a

X(s, γ)∆s, t ∈ [a, b]T

is a continuous time scale uncertain process.

Proof. From Proposition 4.2.5, it follows that the function t 7→∫ taX(s, γ)∆s is

continuous for each γ ∈ Γ. Since the Riemann ∆-integral is a limit of the fi-nite sum S (γ) =

∑ni=1(ti − ti−1)X(ξi, γ) of measurable functions, we have that

γ 7→∫ taX(s, γ)∆s is a measurable function. Therefore, Y (·, ·) is a continuous time

scale uncertain process.

4.3 Uncertain initial value problem

In the following, consider an initial value problem of the formX∆(t, γ) = F (t,X(t, γ), γ), t ∈ [a, b]κT

X(a, γ) = X0(γ),(4.2)

where X0 : Γ → Rm is an uncertain vector and F : [a, b]κT × Rm × Γ → Rm satisfies

the following assumptions:

(H1) F (t, x, ·) : Γ→ Rm is an uncertain variable for all (t, x) ∈ [a, b]κT × Rm,

(H2) for each γ ∈ Γ, the function F (·, ·, γ) : [a, b]κT×Rm → Rm is a Hilger continuous

function at every point (t, x) ∈ [a, b]κT × Rm.

By a solution of (4.2) we mean a time scale uncertain process X(·, ·) : [a, b]κT×Γ→

Rm that satisfies conditions in (4.2).

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Remark 5. Consider the uncertain differential equation (4.2) as a family (with re-spect to parameter γ) of deterministic differential equations, namely

X∆(t, γ) = F (t,X(t, γ), γ), t ∈ [a, b]κT, γ ∈ Γ,X(a, γ) = X0(γ).

(4.3)

Then it is not correct to solve each problem (4.3) to obtain the solutions of (4.2). Letus give two examples.

Example 4.3.1. Let (Γ,L,M) be an uncertain space. Consider an initial valueproblem of the form

X∆(t, γ) = K(γ)X2(t, γ), t ∈ [0,∞)R, γ ∈ Γ,X(0, γ) = 1,

(4.4)

where K : Γ→ (0,∞) is an uncertain variable. It is easy to see that, for each γ ∈ Γ,X(t, γ) = 1

1−K(γ)tis a solution of (4.4) on the interval [0, 1/K(γ)]. Since for each

a ≥ 0 we have that M(1/K(γ) > a) < 1, it follows that not all solutions X(·, γ) arewell defined on some common interval [0, a).

Example 4.3.2. Let (Γ,L,M) be an uncertain space and let Γ0 /∈ L. It is easy tocheck that, for each γ ∈ Γ, the function X(·, ·) : [0, 1]R × Γ→ R, given by

X(t, γ) =

0 if γ ∈ Γ0

t3/2 if γ ∈ Γr Γ0,

is a solution of the initial value problemX∆(t, γ) = 3

2X(t, γ), t ∈ [0,∞)R, γ ∈ Γ,

X(0, γ) = 0.

But X(·, ·) is not an uncertain process. Indeed, we have that

γ ∈ Γ;X(1, γ) ∈ [−1

2,1

2] = Γ0 /∈ L,

that is, γ 7→ X(1, γ) is not a measurable function.

Using the Propositions 4.2.4, 4.2.5 and [100, 101, Lemma 2.3], it is easy to prove

the following result.

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Lemma 4.3.1. A time scale uncertain process X(·, ·) : [a, b]κT × Γ → Rm is thesolution of the problem (4.2) if and only if X(·, ·) is a continuous time scale uncertainprocess and it satisfies the following uncertain integral equation

X(t, γ) = X0(γ) +

∫ t

a

F (s,X(s, γ), γ)∆s, t ∈ [a, b]T, γ ∈ Γ. (4.5)

The following results is known as Gronwall’s inequality on time scale and will be usedin this paper.

Lemma 4.3.2. ([141, Lemma 3.1]). Let rd-continuous time scale uncertain processesX(·, ·), Y (·, ·) : [a, b]κT × Γ→ R+ be such that

X(t, γ) ≤ Y (t, γ) +

∫ t

a

q(s)X(s, γ)∆s, t ∈ [a, b]T, γ ∈ Γ,

where 1 + µ(t)p(t) > 0, for all t ∈ [a, b]T. Then we have

X(t, γ) ≤ Y (t, γ) +

∫ t

a

ep(t, σ(s))p(s)Y (s, γ)∆s, t ∈ [a, b]T, γ ∈ Γ.

Theorem 4.1. Let F : [a, b]κT × Rm × Γ→ Rm satisfies (H1)-(H2) and assume thatthere exists an rd-continuous time scale uncertain process L(·, ·) : [a, b]κT × Γ → R+

such that‖F (t, x, γ)− F (t, y, γ)‖ ≤ L(t, γ) ‖x− y‖ (4.6)

for every t ∈ [a, b]κT, x, y ∈ Rm and γ ∈ Γ. Let X0 : Γ→ Rm an uncertain vector suchthat

‖F (t,X0(γ), γ)‖ ≤M , t ∈ [a, b]κT, γ ∈ Γ, (4.7)

where M > 0 is a constant. Then the problem (4.2) has a unique solution.

Proof. To prove the theorem we apply the method of successive approximations (see[141]). For this, we define a sequence of functions Xn(·, ·) : [a, b]κT × Γ→ Rm, n ∈ N,as follows:

X0(t, γ) = X0(γ)

Xn(t, γ) = X0(γ) +t∫a

F (s,Xn−1(s, γ), γ)∆s, n ≥ 1,(4.8)

for every t ∈ [a, b]κT and every γ ∈ Γ. First, using (4.7) and the Lemma 4.2.1, weobserve that

‖X1(t, γ)−X0(t, γ)‖ ≤t∫

a

‖F (s,X0(γ), γ)‖∆s ≤M(t− a)

≤ M(b− a), t ∈ [a, b]T, γ ∈ Γ.

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We prove by induction that for each integer n ≥ 2 the following estimate holds

‖Xn(t, γ)−Xn−1(t, γ)‖ ≤ML(γ)(t− a)n

n!≤ML(γ)

(b− a)n

n!, t ∈ [a, b]T, γ ∈ Γ,

(4.9)

where L(γ) = sup[a,b]T

L(t, γ). Suppose that (4.9) holds for n = k ≥ 2. Then, using (4.6),

(4.7) and Lemma 4.2.1, we obtain

‖Xk+1(t, γ)−Xk(t, γ)‖ ≤t∫a

‖F (s,Xk(s, γ), γ)− F (s,Xk−1(s, γ), γ)‖∆s

≤ L(γ)t∫a

‖Xk(s, γ)−Xk−1(s, γ)‖∆s ≤ L(γ)Mk!

t∫a

(s− a)k∆s

≤ L(γ)Mk!

t∫a

(s− a)kds = ML(γ) (t−a)k+1

(k+1)!≤ML(γ) (b−a)k+1

(k+1)!,

for all t ∈ [a, b]T and γ ∈ Γ. Thus, (4.9) is true for n = k + 1 and so (4.9) holds forall n ≥ 2. Further, we show that for every n ∈ N the functions Xn(·, γ) : [a, b]T → Rare continuous for each γ ∈ Γ. Let ε > 0 and t, s ∈ [a, b]T be such that |t− s| < ε/M .We have

‖X1(t, γ)−X1(s, γ)‖ =

∥∥∥∥ t∫a

F (τ,X0(γ), γ)∆τ −s∫a

F (τ,X0(γ), γ)∆τ

∥∥∥∥=

∥∥∥∥ t∫s

F (τ,X0(γ), γ)∆τ

∥∥∥∥ ≤ t∫s

‖F (τ,X0(γ), γ)‖∆τ ≤t∫s

‖F (τ,X0(γ), γ)‖ dτ

≤M |t− s| < ε

and so t 7→ X1(t, γ) is continuous on [a, b]T for each γ ∈ Γ. Since for each n ≥ 2

‖Xn(t, γ)−Xn(s, γ)‖ ≤t∫s

‖F (τ,Xn−1(τ, γ), γ)‖∆τ ≤t∫s

‖F (τ,X0(γ), γ)‖∆τ

+t∫s

‖F (τ,Xn−1(τ, γ), γ)− F (τ,X0(γ), γ)‖∆τ ≤t∫s

‖F (τ,X0(γ), γ)‖∆τ

+n−1∑k=1

t∫s

‖F (τ,Xk(τ, γ), γ)− F (τ,Xk−1(τ, γ), γ)‖∆τ

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then, by induction, we obtain

‖Xn(t, γ)−Xn(s, γ)‖ ≤M

(1 +

n−1∑k=1

L(γ)k−1(b−a)k

k!

)|t− s| → 0 as s→ t.

Therefore, for every n ∈ N the function Xn(·, γ) : [a, b]T × Γ → Rm is continuousfor each γ ∈ Γ. Now, using Lemma 4.3.1 and (4.8), we deduce that the functionsXn(t, ·) : Γ → Rm are measurable. Consequently, it follows that for every n ∈ N thefunction Xn(·, ·) : [a, b]T × Γ→ R is a time scale uncertain process.

Further, we shall show that the sequence (Xn(t, ·))n∈N is uniformly convergent.Denote

Yn(t, γ) = ‖Xn+1(t, γ)−Xn(t, γ)‖ , n ∈ N, γ ∈ Γ.

Since

Yn(t, γ)− Yn(s, γ) ≤ L(γ)

t∫s

‖Xn(τ, γ)−Xn−1(τ, γ)‖∆τ

then, reasoning as above, we deduce that the functions t 7→ Yn(t, γ) are continuouson [a, b]T for each γ ∈ Γ. Now, using (4.9), we obtain

supt∈[a,b]T

‖Xn(t, γ)−Xm(t, γ)‖ ≤n−1∑k=m

supt∈[a,b]T

Yk(t, γ) ≤Mn−1∑k=m

L(γ)k(b− a)k+1

(k + 1)!

for all n > m > 0. Since the series∞∑n=1

L(γ)n−1(b − a)n/n! converges, then for each

ε > 0 there exists n0 ∈ N such that

supt∈[a,b]T

‖Xn(t, γ)−Xm(t, γ)‖ ≤ ε for all n,m ≥ n0 and γ ∈ Γ. (4.10)

Hence, since ([a, b]T, | · |) is a complete metric space, it follows that the sequence(Xn(t, ·))n∈N is uniformly convergent on [a, b]T. Denote X(t, γ) = lim

n→∞Xn(t, γ), t ∈

[a, b]T, γ ∈ Γ. Obviously, t 7→ X(t, γ) is continuous on [a, b]T for each γ ∈ Γ.Since, by Lemma 4.2.2 and (4.8), the functions γ → Xn(·, γ) are measurable andX(t, γ) = lim

n→∞Xn(t, γ) for every t ∈ [a, b]T and γ ∈ Γ, we deduce that γ → X(t, γ) is

measurable for every t ∈ [a, b]T. Therefore, X(·, ·) : [a, b]T × Γ→ Rm is a continuoustime scale uncertain process. We show that X(·, ·) satisfies the uncertain integralequation (4.5). For each n ∈ N we put Gn(t, γ) = F (t,Xn(t, γ), γ), t ∈ [a, b]T, γ ∈ Γ.Then Gn(t, γ) is rd-continuous time scale uncertain process, and we have that

supt∈[a,b]T

‖Gn(t, γ)−Gm(t, γ)‖ ≤ L(γ) supt∈[a,b]T

‖Xn(t, γ)−Xm(t, γ)‖ , t ∈ [a, b]T, γ ∈ Γ,

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for all n,m ≥ n0. Using (4.10) we infer that the sequence (Gn(·, γ))n∈N is uniformlyconvergent on [a, b]T for each γ ∈ Γ. If we take m → ∞, then for each ε > 0 thereexists n0 ∈ N such that for every n ≥ n0 we have

supt∈[a,b]T

‖Gn(t, γ)− F (t,X(t, γ), γ)‖ ≤ L(γ) supt∈[a,b]T

‖Xn(t, γ)−X(t, γ)‖ , t ∈ [a, b]T, γ ∈ Γ,

and so limn→∞

‖Gn(t, γ)− F (t,X(t, γ), γ)‖ = 0 for all t ∈ [a, b]T and γ ∈ Γ. Also, it easy

to see that

supt∈[a,b]T

∥∥∥∥∥∥t∫

a

Gn(s, γ)∆s−t∫

a

F (s,X(s, γ), γ)∆s

∥∥∥∥∥∥ ≤ L(γ)

t∫a

‖Xn(s, γ)−X(s, γ)‖∆s, γ ∈ Γ.

Since X(t, γ) = limn→∞

Xn(t, γ) uniformly on [a, b]T, then it follows that

limn→∞

t∫a

Gn(s, γ)∆s =

t∫a

F (s,X(s, γ), γ)∆s for all t ∈ [a, b] and γ ∈ Γ.

Now, we have

supt∈[a,b]T

∥∥∥∥X(t, γ)−X0(γ)−t∫a

F (s,X(s, γ), γ)∆s

∥∥∥∥ ≤ supt∈[a,b]T

‖X(t, γ)−Xn(t, γ)‖

+ supt∈[a,b]T

∥∥∥∥Xn(t, γ)−X0(γ)−t∫a

F (s,Xn−1(s, γ), γ)∆s

∥∥∥∥+ supt∈[a,b]T

∥∥∥∥ t∫a

F (s,Xn−1(s, γ), γ)∆s−t∫a

F (s,X(s, γ), γ)∆s

∥∥∥∥ .Using the two previous convergences

X(t, γ) = X0(γ) +

t∫a

F (s,X(s, γ), γ)∆s for all t ∈ [a, b]T and γ ∈ Γ;

that is, X(·, ·) satisfies the uncertain integral equation (4.5). Then, by Lemma 4.3.2,it follows that X(·, ·) is the solution of the problem (4.2). Finally, we show theuniqueness of the solution. For this, we assume that X(·, ·), Y (·, ·) : [a, b]T× Γ→ Rmare two solutions of (4.5). Since

‖X(t, γ)− Y (t, γ)‖ ≤∫ t

a

L(γ) ‖X(s, γ)− Y (s, γ)‖ ds, t ∈ [a, b]T, γ ∈ Γ,

from Lemma 3.4 it follows that ‖X(t, γ)− Y (t, γ)‖ ≤ 0, t ∈ [a, b]T, γ ∈ Γ, and so,the proof is complete.

63

Let T be an upper unbounded time scale. Then under suitable conditions we can

extend the notion of the solution of (4.2) from [a, b]κT to [a,∞)T := [a,∞) ∩ T, if we

define F on [a,∞)T×Rm× Γ and show that the solution exists on each [a, b]T where

b ∈ (a,∞)T, a < ρ(b).

Theorem 4.2. Assume that F : [a,∞)T × Rm × Γ → Rm satisfies the assumptionsof the Theorem 3.1 on each interval [a, b]T with b ∈ (a,∞)T, a < ρ(b). If there is aconstant M > 0 such that ‖F (t, x, γ)‖ ≤ M for all (t, x) ∈ [a, b)T × Rm then theproblem (4.2) has a unique solution on [a,∞)T.

Proof. Let X(·, ·) be the solution of (4.2) which exists on [a, b)T with b ∈ (a,∞)T,a < ρ(b), and the value of b cannot be increased. First, we observe that b is a left-scattered point, then ρ(b) ∈ (a, b)T and the solution X(·, ·) exists on [a, ρ(b)]T. Butthen the solution X(·, ·) exists also on [a, b]T, namely by putting

X(b, γ) = X(ρ(b), γ) + µ(b)X∆(ρ(b), γ)

= X(ρ(b), γ) + µ(b)F (ρ(b), X(ρ(b), γ), γ).

If b is a left-dense point, then their neighborhoods contain infinitely many points tothe left of b. Then, for any t, s ∈ (a, b)T such that s < t, we have

‖X(t, γ)−X(s, γ)‖ ≤∫ t

s

‖F (τ,X(τ, γ), γ)‖∆τ ≤M |t− s| .

Taking limit as s, t → b− and using Cauchy criterion for convergence, it followslimt→b−

X(t, γ) exists and is finite. Further, we define Xb(γ) = limt→b−

X(t, γ) and consider

the initial value problemX∆(t, γ) = F (τ,X(τ, γ), γ), t ∈ [b, b1]T, b1 > σ(b),X(b, γ) = Xb(γ).

By Theorem 3.1, one gets that X(t, γ) can be continued beyond b, contradicting ourassumptions. Hence every solution X(t, γ) of (4.2) exists on [a,∞)T and the proof iscomplete.

4.4 Uncertain linear systems

Let a : Γ→ R be a positively regressive uncertain variable; that is, 1 + µ(t)a(γ) > 0

for all γ ∈ Γ. Then, by Lemma 2.2, the function (t, γ) 7→ ea(γ)(t, t0) defined by

ea(γ)(t, t0) =

(∫ t

t0

log(1 + µ(τ)a(γ))

µ(τ)∆τ

), t0, t ∈ T, γ ∈ Γ,

64

is a continuous time scale uncertain process. For each fixed γ ∈ Γ, the sample path

t 7→ ea(γ)(t, t0) is the exponential function on time scales (see [31]). It easy to check

that the uncertain process (t, γ) 7→ ea(γ)(t, t0) is a solution of the initial value problem

(for deterministic case, see [31, Theorem 2.33]):X∆(t, γ) = a(γ)X(t, γ), t ∈ [t0, b]

κT, γ ∈ Γ,

X(t0, γ) = 1.(4.11)

If a : Γ→ R is bounded then, by the Theorems 4.1 and 4.2, it follows that (4.11) has

a unique solution on [t0,∞)T.

Let us denote by Mm(R) the space of all m×m matrices. We recall that ||A|| :=

sup||Ax||; ||x|| ≤ 1 define a norm on Mm(R) and the following inequality ||Ax|| ≤

||A|| · ||x|| holds for all A ∈ Mm(R) and x ∈ Mm(R)m. A mapping A : Γ→ Mm(R)

is called an uncertain matrix if all its components aij : Γ → R, i, j = 1, 2, ...,m, are

uncertain variables. An uncertain matrix A is said to be regressive if I + µ(t)A(γ) is

invertible for all t ∈ T and γ ∈ Γ, where I is the m×m identity matrix. Moreover, the

set Rm = R(Γ,Mm(R)) of all regressive uncertain matrices is a group with respect

to the addition operation ⊕ define

A⊕B = A+B + µ(t)AB

for all t ∈ T. The inverse element of A ∈ Rm is given by

A = −[I + µ(t)A−1]A = −A[I + µ(t)A]−1

for all t ∈ T.

Now consider the following homogeneous linear uncertain initial value problem

X∆(t, γ) = A(γ)X(t, γ), t ∈ T, γ ∈ Γ,

X(t0, γ) = X0(γ).(4.12)

65

where A ∈ Rm. The corresponding nonhomogeneous linear uncertain initial value

problem is

X∆(t, γ) = A(γ)X(t, γ) +H(t, γ), t ∈ T, γ ∈ Γ,

X(t0, γ) = X0(γ).(4.13)

where H : T× Γ→ Rm is an uncertain process.

Theorem 4.3. Suppose that A : Γ → Mm(R) is a regressive and bounded uncertainmatrix, X0 : Γ→ Rm is a bounded uncertain variable, and H(·, ·) : [t0,∞)T×Γ→ Rmis a rd-continuous time scale uncertain process. If there is a constant ν > 0 such that‖H(t, γ)‖ ≤ ν for all t ∈ [t0, b)T with b ∈ (t0,∞)T, t0 < ρ(b), then the initial valueproblem (4.13) has a unique solution on [t0,∞)T.

Proof. First, we observe that we put F (t, x, γ) := A(γ)x + H(t, γ), then F satisfiesthe conditions (H1) and (H2). Moreover,

‖F (t, x, γ)− F (t, y, γ)‖ ≤ ‖A(γ)‖ ‖x− y‖

for every t ∈ [t0,∞)T, x, y ∈ Rm and γ ∈ Γ. Therefore, by the Theorem 4.1, it followsthat (4.13) has a unique solution on [t0, b]

κT. Further, let X(t, ·) be the solution of

(4.13) which exists on [t0, b)T with b ∈ (t0,∞)T, t0 < ρ(b). Also, let N > 0 be suchthat ‖A(γ)‖ ≤ N . Then we have

‖X(t, γ)‖ ≤ ‖X(t0, γ)‖+

∫ t

t0

‖A(γ)X(s, γ)‖∆s+

∫ t

t0

‖H(s, γ)‖∆s ≤

1 + ν(t− t0) +N

∫ t

t0

‖X(s, γ)‖∆s.

Then, by the Corollary 6.8 in [31], it follows that

‖X(t, γ)‖ ≤ (1 +ν

N)eN(t, t0)− ν

N≤ (1 +

ν

N)eN(b, t0).

Hence ‖F (t,X(t, γ), γ)‖ ≤ M := ν + (1 + νN

)eN(b, t0). Proceeding as in the proof ofthe Theorem 4.2 it follows that the unique solution of (4.13) exists on [t0,∞)T.

A mapping Ψ : T × Γ → Mm(R) is called an uncertain matrix process if all its

components ψij : T × Γ → R, i, j = 1, 2, ...,m, are uncertain process. An uncertain

matrix process Ψ is said to be regressive if Ψ(t, ·) ∈ Rm for all t ∈ T. In the following,

suppose that A : Γ → Mm(R) is a regressive and bounded uncertain matrix. An

66

uncertain matrix process ΨA is said to be an uncertain matrix solution of the the

following homogeneous linear uncertain differential equation

X∆(t, γ) = A(γ)X(t, γ), t ∈ T, γ ∈ Γ, (4.14)

if each column of ΨA satisfies (4.14). An uncertain fundamental matrix of (4.14) is

an uncertain matrix solution ΨA of (4.14) such that det ΨA(t, γ) 6= 0 for all t ∈ T and

γ ∈ Γ. An uncertain transition matrix of (4.14) at initial time s ∈ T is an uncertain

fundamental matrix ΨA such that ΨA(s, γ) = I for all γ ∈ Γ. The uncertain transition

matrix of (4.14) at initial time s ∈ T will be denoted by UA(t, s). Therefore, the

uncertain transition matrix of (4.14) at initial time s ∈ T is the unique solution of

the following uncertain matrix initial value problem

Φ∆(t, γ) = A(γ)Φ(t, γ), Φ(s, γ) = I, (4.15)

and X(t, γ) = UA(t, s)X(s, γ), t ≥ s, is the unique solution of the following uncertain

initial value problem

X∆(t, γ) = A(γ)X(t, γ), t ∈ T, γ ∈ Γ,

X(s, γ) = X0(γ).

The existence and uniqueness of the solution of (4.15) follows from the Theorem

4.1. The uncertain transition matrix of (4.14) at initial time s ∈ T is also called

the uncertain matrix exponential function (at s), and it is denoted by eA(γ)(t, s) or

eA(t, s) .

In the following theorem we give some properties of the uncertain transition ma-

trix. The proof of the theorem is the same that in [31, Theorem 5.21].

Theorem 4.4. If A : Γ→Mm(R) is a regressive and bounded uncertain matrix, then

67

(1) UA(t, t) = I;

(2) UA(σ(t), s) = [I + µ(t)A(γ)]UA(t, s);

(3) U−1A (t, s) = UT

AT (t, s);

(4) UA(t, s) = U−1A (s, t) = UT

AT (s, t);

(5) UA(t, s)UA(s, r) = UA(t, r),

for all t, s, r ∈ T with t > s > r and all γ ∈ Γ.

Theorem 4.5. (Variation of Constants). Suppose that the assumptions of the The-orem 4.3 hold. Then the unique solution X(·, ·) : [t0,∞)T× Γ→ Rm of the uncertaininitial value problem (4.13) is given by

X(t, γ) = UA(t, t0)X0(γ) +

∫ t

t0

UA(t, σ(s))H(s, γ)∆s, t ∈ [t0,∞)T, γ ∈ Γ. (4.16)

Proof. Indeed, we can rewritten (4.16) as

X(t, γ) = UA(t, t0)

[X0(γ) +

∫ t

t0

UA(t0, σ(s))H(s, γ)∆s

].

Using the product rule to differentiate X(t, ·), we infer

X∆(t, γ) = A(γ)UA(t, t0)

[X0(γ) +

∫ t

t0

UA(t0, σ(s))H(s, γ)∆s

]+UA(σ(t), t0)UA(t0, σ(t))H(t, γ)

= A(γ)X(t, γ) +H(t, γ)

Obviously, X(t0, γ) = X0(γ). Therefore, X(t, γ) is the solution of (4.16).

Corollary 4.4.1. Let X0 : Γ → Rm be a bounded uncertain variable. If A : Γ →Mm(R) is a regressive and bounded uncertain matrix, then the unique solution of theuncertain initial value problem (4.12) is given by

X(t, γ) = UA(t, t0)X0(γ), t ∈ [t0,∞)T, γ ∈ Γ.

Theorem 4.6. (Variation of Constants). Suppose that the assumptions of the The-orem 4.3 hold. Then the unique solution X(·, ·) : [t0,∞)T × Γ→ Rm of the followinguncertain initial value problem

X∆(t, γ) = −AT (γ)Xσ(t, γ) +H(t, γ), t ∈ [t0,∞)T, γ ∈ Γ,X(t0, γ) = X0(γ),

(4.17)

68

on [t0,∞)T given by

X(t, γ) = ΨAT (γ)(t, t0)X0(γ) +

∫ t

t0

ΨAT (γ)(t, s)H(s, γ)∆s, t ∈ [t0,∞)T, γ ∈ Γ.

(4.18)

Proof. . Indeed, we can rewrite (4.17) as

X∆(t, γ) = −AT (γ)[X(t, γ) + µ(t)X∆(t, γ)] +H(t, γ)

= −AT (γ)X(t, γ)− µ(t)AT (γ)X∆(t, γ) +H(t, γ);

that is,[I + µ(t)AT (γ)]X∆(t, γ) = −AT (γ)X(t, γ) +H(t, γ).

Since the matrix A(γ) is regressive, then AT (γ) is also regressive, and hence we inferthat

X∆(t, γ) = −[I + µ(t)AT (γ)]−1AT (γ)X(t, γ) + [I + µ(t)AT (γ)]−1H(t, γ)

= AT (γ)X(t, γ) + [I + µ(t)AT (γ)]−1H(t, γ);

that is,X∆(t, γ) = AT (γ)X(t, γ) + [I + µ(t)AT (γ)]−1H(t, γ).

Now, using the Theorem 4.3 and the properties of the uncertain transition matrix,we obtain that

X(t, γ) = UAT (t, t0)X0(γ) +

∫ t

t0

UAT (t, σ(s))[I + µ(t)AT (γ)]−1H(s, γ)∆s

= UAT (t, t0)X0(γ) +

∫ t

t0

UTA (t, σ(s))[I + µ(t)AT (γ)]−1H(s, γ)∆s

= UAT (t, t0)X0(γ) +

∫ t

t0

[I + µ(t)A(γ)]−1UA(σ(s), t)TH(s, γ)∆s

= UAT (t, t0)X0(γ) +

∫ t

t0

UA(s, t)H(s, γ)∆s;

that is, (4.18).

Corollary 4.4.2. Let X0 : Γ→ R be a bounded uncertain variable. If A : Γ→Mm(R)is a regressive and bounded uncertain matrix, then the unique solution of the followinguncertain initial value problem

X∆(t, γ) = −A(γ)Xσ(t, γ), t ∈ [t0,∞)T, γ ∈ Γ,X(t0, γ) = X0(γ),

is given byX(t, γ) = UAT (t, t0)X0(γ), t ∈ [t0,∞)T, γ ∈ Γ.

69

Example 4.4.1. Let us consider Γ = (0, 1), L the σ-algebra of all Borel subsets ofΓ, M the uncertain measure on Γ defined in Example 2.1, and the following initialvalue problem

X∆(t, γ) = γX(t, γ) + eγ(t, 0), t ∈ [0,∞)T, γ ∈ Γ,X(0, γ) = γ.

(4.19)

Then, by the Theorems 4.3 and 4.4, the initial value problem (4.19) has a uniquesolution on [0,∞)T, given by

X(t, γ) = γeγ(t, 0) +

∫ t

0

eγ(t, σ(s))eγ(s, 0)∆s;

that is,

X(t, γ) = eγ(t, 0)

(γ +

∫ t

0

1

1 + µ(s)γ∆s

), t ∈ [0,∞)T.

Next, consider two particular cases.

If T = R, then µ(t) = 0 for all t ∈ N, and eγ(t, 0) = eγt. Moreover, in this casewe have ∫ t

0

1

1 + µ(s)γ∆s =

∫ t

0

ds = t.

It follows that the initial value problemX∆(t, γ) = γX(t, γ) + eγt, t ∈ [0,∞)X(0, γ) = γ,

has the solution X(t, γ) = (γ + t)eγt, t ∈ [0,∞).

If T = N, then µ(n) = 1 for all n ∈ N, and eγ(n, 0) = (1 + γ)n. Moreover, in thiscase we have ∫ t

0

1

1 + µ(s)γ∆s =

∑s∈[0,n)

1

1 + γ=

n

1 + γ.

It follows that the difference initial value problemXn+1(γ) = (1 + γ)Xn(γ) + (1 + γ)n, n ∈ NX0(γ) = γ,

has the solution Xn(γ) = (γ + n1+γ

)(1 + γ)n, n ∈ N.

70

Example 4.4.2. Let us consider Γ = (0, 1), L the σ-algebra of all Borel subsets ofΓ, M the uncertain measure on Γ defined in Example 4.2.1, and the following initialvalue problem

X∆(t, γ) = −γXσ(t, γ) + eγ(t, t0), t ∈ [0,∞)T, γ ∈ Γ,X(0, γ) = γ.

(4.20)

The initial value problem (4.20) has a unique solution on [t0,∞)T, given by

X(t, γ) = γeγ(t, t0) +

∫ t

0

eγ(t, s)eγ(s, 0)∆s;

that is,X(t, γ) = (γ + t) eγ(t, 0), t ∈ [0,∞)T, γ ∈ (0, 1).

If T = R, then µ(t) = 0 for all t ∈ R, and eγ(t, 0) = e−γt. It follows that X(t, γ) =(γ + t) = e−γt, t ∈ [0,∞)T, γ ∈ (0, 1).

If T = hN with h > 0, then µ(t) = h for all t ∈ hN, and eγ(t, 0) = (1 + γh)−t/h.It follows that the h-difference initial value problem

Xt+h(γ) = 11+γh

Xt(γ) + h(1 + γh)−t/h−1, t ∈ hNX0(γ) = γ,

has the unique solution Xt(γ) = (γ + t) (1 + γh)−t/h, t ∈ hN.

If T = 2N, then µ(t) = t for all t ∈ 2N, and eγ(t, 0) =∏

s∈[0,t)

(1 + γs)−1. It follows

that the 2-difference initial value problemXt(γ) = (1 + γt)X2t(γ)− t

∏s∈[1,t)

(1 + γs)−1, t ∈ 2N

X1(γ) = γ,

has the unique solution Xt(γ) = (γ + t)∏

s∈[1,t)

(1 + γs)−1, t ∈ 2N.

Example 4.4.3. Let us consider Γ = (0, 1), L the σ-algebra of all Borel subsets ofΓ, M the uncertain measure on Γ defined in Example 4.2.1, and the following initialvalue problem

X∆(t, γ) =

[1 γ0 −1

]X(t, γ), X(0, γ) =

[γ1

], t ∈ [0,∞)T, (4.21)

where 1− µ(t) 6= 0 for t ∈ [0,∞)T. The matrix A =

[1 γ0 −1

]has the eingenvalues

λ1 = −1, λ2 = 1 with the corresponding eigenvectors v1 =

[10

], v1 =

[1−γ

2

],

71

respectively. Then

ΨA(t, γ) =

[e−1(t, 0) e1(t, 0)

0 − 2γe1(t, 0)

], Ψ−1

A (s, γ) =

[e1(s, 0) γ

2e1(s, 0)

0 γ2e−1(s, 0)

]and therefore, the uncertain transition matrix for (4.21) is given by

UA(t, s) =

[e−1(t, s) γ

2e−1(t, s)− γ

2e1(t, s)

0 e−1(t, s)

].

It follows that the solution of the initial value problem (4.21) is given by

X(t, γ) = UA(t, 0)X(0, γ) =

[3γ2e−1(t, 0)− γ

2e1(t, 0)

e−1(t, 0)

], t ∈ [0,∞)T, γ ∈ (0, 1).

Conclusion

In this we investigated certain classes of differential equations that arise in the model-

ing of the dynamic systems with uncertainty as: set differential equations with causal

operators, neutral set differential equations, fuzzy delay differential equations and

uncertain dynamic systems on time scales. For each type of differential equation,

we established the existence and/or uniqueness of the solutions. Also, we gave some

results on the continuity of the solutions with respect to the initial data. Investigating

these types of equations is very important in the modeling of dynamic processes with

uncertainty. For our future research we intend to study the set differential equations

driven by processes Liu, set differential equations with fuzzy or uncertain parame-

ters, fuzzy differential equations on time scales, and their applications in biology or

finance.

72

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